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Elementary Algebra

302 questions

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Question 121Question →

Which of the following is a factor of the polynomial 4x2−12xy+9y2−254x^2 - 12xy + 9y^2 - 25?

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Answer: 2x−3y−52x - 3y - 5

Answer

The expression 2x−3y−52x - 3y - 5 is a factor of the polynomial.
The polynomial can be factored by first grouping the first three terms as a perfect square trinomial: 4x2−12xy+9y2=(2x−3y)24x^2 - 12xy + 9y^2 = (2x - 3y)^2. This simplifies the expression to (2x−3y)2−25(2x - 3y)^2 - 25. Since 25=5225 = 5^2, this is a difference of squares of the form A2−B2A^2 - B^2, which factors into (A−B)(A+B)(A - B)(A + B). Substituting A=2x−3yA = 2x - 3y and B=5B = 5 gives the factored form (2x−3y−5)(2x−3y+5)(2x - 3y - 5)(2x - 3y + 5). Thus, the option representing 2x−3y−52x - 3y - 5 is a factor.

Step-by-Step Solution

1
Group the first three terms of the polynomial and recognize the perfect square trinomial pattern.
4x2−12xy+9y2=(2x−3y)24x^2 - 12xy + 9y^2 = (2x - 3y)^2
The term 4x24x^2 is (2x)2(2x)^2, 9y29y^2 is (3y)2(3y)^2, and the middle term −12xy-12xy is −2(2x)(3y)-2(2x)(3y), which matches the perfect square trinomial identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
2
Rewrite the original expression using the factored trinomial and write 25 as a perfect square.
(2x−3y)2−52(2x - 3y)^2 - 5^2
Substituting the factored trinomial and expressing 25 as 525^2 sets up the expression as a difference of squares in the form A2−B2A^2 - B^2.
3
Apply the difference of squares factoring formula A2−B2=(A−B)(A+B)A^2 - B^2 = (A - B)(A + B).
(2x−3y−5)(2x−3y+5)(2x - 3y - 5)(2x - 3y + 5)
Substituting A=2x−3yA = 2x - 3y and B=5B = 5 into the formula yields the completely factored polynomial.

Key Concept

Factoring by grouping using perfect square trinomials and difference of squares
Estimated Time:1m 0s
Question 122Question →

For all real numbers uu and vv, which of the following is equivalent to the expression 12(2u−3v)2−23u(3u−9v)−23v2\frac{1}{2}(2u - 3v)^2 - \frac{2}{3}u(3u - 9v) - \frac{2}{3}v^2?

Show answer & explanation

Answer: 236v2\frac{23}{6}v^2

Answer

The simplified equivalent expression is 236v2\frac{23}{6}v^2.
The correct answer is obtained by expanding (2u−3v)2(2u - 3v)^2 to 4u2−12uv+9v24u^2 - 12uv + 9v^2, multiplying it by 12\frac{1}{2} to get 2u2−6uv+92v22u^2 - 6uv + \frac{9}{2}v^2, distributing −23u-\frac{2}{3}u to get −2u2+6uv-2u^2 + 6uv, and combining the terms: (2u2−2u2)+(−6uv+6uv)+(92−23)v2=236v2(2u^2 - 2u^2) + (-6uv + 6uv) + (\frac{9}{2} - \frac{2}{3})v^2 = \frac{23}{6}v^2.

Step-by-Step Solution

1
Expand the squared binomial (2u−3v)2(2u - 3v)^2.
4u2−12uv+9v24u^2 - 12uv + 9v^2
Using the binomial square formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 allows us to expand the expression before applying the outer coefficient.
2
Multiply the expanded binomial by the coefficient 12\frac{1}{2}.
2u2−6uv+92v22u^2 - 6uv + \frac{9}{2}v^2
Distributing the constant factor of 12\frac{1}{2} to each term of the expanded binomial.
3
Distribute the term −23u-\frac{2}{3}u across the parenthetical expression (3u−9v)(3u - 9v).
−2u2+6uv-2u^2 + 6uv
Multiplying each term inside the parentheses by −23u-\frac{2}{3}u, making sure to distribute the negative sign properly: −23u⋅3u=−2u2-\frac{2}{3}u \cdot 3u = -2u^2 and −23u⋅(−9v)=6uv-\frac{2}{3}u \cdot (-9v) = 6uv.
4
Combine all terms and group like terms.
2u2−6uv+92v2−2u2+6uv−23v22u^2 - 6uv + \frac{9}{2}v^2 - 2u^2 + 6uv - \frac{2}{3}v^2
Write the full expression with all distributed terms to identify and combine like terms.
5
Combine the u2u^2, uvuv, and v2v^2 terms.
236v2\frac{23}{6}v^2
Combining the coefficients: 2u2−2u2=02u^2 - 2u^2 = 0, −6uv+6uv=0-6uv + 6uv = 0, and 92v2−23v2=(276−46)v2=236v2\frac{9}{2}v^2 - \frac{2}{3}v^2 = (\frac{27}{6} - \frac{4}{6})v^2 = \frac{23}{6}v^2.

Key Concept

Simplifying expressions by expanding binomials, distributing negative signs, and combining like terms with fractional coefficients.

Alternative Method

Instead of algebraic expansion, you can substitute simple non-zero values for uu and vv (e.g., u=3u = 3 and v=2v = 2) into the original expression and evaluate it. Then, substitute the same values into the answer choices to find which one yields the same result.
Estimated Time:1m 30s
Question 123Question →

A newly designed temperature scale, Scale X, is related to the Celsius scale (∘C^{\circ}\text{C}) by a linear equation. Water freezes at 0∘C0^{\circ}\text{C}, which corresponds to −15∘X-15^{\circ}\text{X}, and water boils at 100∘C100^{\circ}\text{C}, which corresponds to 135∘X135^{\circ}\text{X}. If a chemical reaction must be maintained at a temperature where the reading on Scale X is exactly 2.52.5 times the reading on the Celsius scale, what is this temperature in degrees Celsius?

Show answer & explanation

Answer: −15.0-15.0

Answer

−15.0-15.0 degrees Celsius
By writing the linear relationship between Scale X (XX) and Celsius (CC) as X=mC+kX = mC + k, we determine the constants using the given coordinates: (0,−15)(0, -15) yields k=−15k = -15, and (100,135)(100, 135) yields m=1.5m = 1.5. The resulting equation is X=1.5C−15X = 1.5C - 15. We then substitute the given condition X=2.5CX = 2.5C, resulting in 2.5C=1.5C−152.5C = 1.5C - 15. Subtracting 1.5C1.5C from both sides gives C=−15C = -15.

Step-by-Step Solution

1
Set up the general linear equation relating Scale X (XX) and Celsius (CC).
X=mC+kX = mC + k
Since the relationship is linear, it can be represented by a slope-intercept linear model.
2
Use the freezing point of water to find the y-intercept kk.
When C=0C = 0, X=−15X = -15, so −15=m(0)+k  ⟹  k=−15-15 = m(0) + k \implies k = -15.
The freezing point of water provides the point (0,−15)(0, -15) on the linear graph.
3
Use the boiling point of water to find the slope mm.
When C=100C = 100, X=135X = 135, so 135=m(100)−15  ⟹  150=100m  ⟹  m=1.5135 = m(100) - 15 \implies 150 = 100m \implies m = 1.5.
The boiling point of water provides the second point (100,135)(100, 135) to determine the rate of change.
4
Substitute the condition X=2.5CX = 2.5C into the linear equation and solve for CC.
2.5C=1.5C−15  ⟹  1.0C=−15  ⟹  C=−152.5C = 1.5C - 15 \implies 1.0C = -15 \implies C = -15.
This isolates the Celsius variable to find the temperature where the Scale X value is exactly 2.52.5 times the Celsius value.

Key Concept

Formulating and solving a linear equation from word-problem constraints and coordinate pairs.

Alternative Method

Instead of deriving the full equation, you can test the options directly. For example, check −15∘C-15^{\circ}\text{C}. The distance from freezing (0∘C0^{\circ}\text{C}) to −15∘C-15^{\circ}\text{C} is −15-15 units. Since Scale X changes by 1.51.5 units for every 11 unit of Celsius (calculated from a change of 150150 on Scale X for 100100 on Celsius), Scale X will change by 1.5×(−15)=−22.51.5 \times (-15) = -22.5 units from its freezing point value of −15-15. This yields −15−22.5=−37.5∘X-15 - 22.5 = -37.5^{\circ}\text{X}. Checking the ratio: −37.5−15=2.5\frac{-37.5}{-15} = 2.5, which matches the given condition.
Estimated Time:2m 0s
Question 124Question →

A craft shop sells handmade candles. The price of a large candle is 33 dollars more than twice the price of a small candle. If a large candle costs 1515 dollars, what is the price, in dollars, of a small candle?

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Answer: 6

Answer

The price of a small candle is 66 dollars.
By letting ss represent the price of a small candle, the price of a large candle is 2s+32s + 3. Since the large candle costs 1515 dollars, we write the equation 2s+3=152s + 3 = 15. Subtracting 33 from both sides gives 2s=122s = 12, and dividing by 22 yields s=6s = 6. Therefore, the price of a small candle is 66 dollars.

Step-by-Step Solution

1
Define the variable and translate the verbal description into an algebraic expression.
Let ss be the price of a small candle. The price of a large candle is expressed as 2s+32s + 3.
Translating 'twice the price of a small candle' to 2s2s and '3 more than' to +3+ 3 allows us to represent the large candle's cost algebraically.
2
Formulate an equation by setting the expression equal to the known cost of the large candle.
2s+3=152s + 3 = 15
The problem states that the large candle costs 1515 dollars.
3
Solve the equation for the variable ss.
s=6s = 6
Subtracting 33 from both sides gives 2s=122s = 12. Dividing both sides by 22 isolates ss, resulting in 66.

Key Concept

Translating verbal statements into linear equations and solving for a single variable.

Alternative Method

We can solve the problem by working backward from the price of the large candle. Since the large candle (1515 dollars) is 33 dollars more than twice the small candle's price, we subtract 33 dollars to find twice the price of the small candle: 15−3=1215 - 3 = 12 dollars. Then, since 1212 dollars is twice the price of the small candle, we divide by 22 to find the price of a single small candle: 12÷2=612 \div 2 = 6 dollars.
Estimated Time:45s
Question 125Question →

What is the maximum integer value of xx that satisfies the inequality 3(2−3x)4−2(2x+5)3≥1−x2+76\frac{3(2 - 3x)}{4} - \frac{2(2x + 5)}{3} \geq \frac{1 - x}{2} + \frac{7}{6}?

Show answer & explanation

Answer: -2

Answer

The maximum integer value of xx that satisfies the inequality is −2-2.
Multiplying the inequality by the common denominator 12 and simplifying yields the inequality −37x≥42-37x \ge 42. Dividing by −37-37 requires reversing the inequality sign, which gives x≤−4237x \le -\frac{42}{37}. The value of −4237-\frac{42}{37} is approximately −1.135-1.135. The largest integer less than or equal to −1.135-1.135 is −2-2.

Step-by-Step Solution

1
Multiply both sides of the inequality by the least common multiple of the denominators (12).
9(2−3x)−8(2x+5)≥6(1−x)+149(2 - 3x) - 8(2x + 5) \geq 6(1 - x) + 14
This eliminates the fractions and simplifies the algebraic manipulation.
2
Expand the terms on both sides of the inequality.
18−27x−16x−40≥6−6x+1418 - 27x - 16x - 40 \geq 6 - 6x + 14
Expanding the terms allows us to combine like terms.
3
Combine the constant and variable terms on each side.
−43x−22≥20−6x-43x - 22 \geq 20 - 6x
This simplifies the inequality to a standard linear form.
4
Add 6x6x and 2222 to both sides to isolate the variable term on the left.
−37x≥42-37x \geq 42
Grouping variable terms on one side and constant terms on the other prepares for the final division.
5
Divide both sides by −37-37 and reverse the direction of the inequality sign.
x≤−4237x \leq -\frac{42}{37}
Dividing an inequality by a negative number requires flipping the inequality sign.
6
Find the largest integer that is less than or equal to −4237-\frac{42}{37}.
−2-2
Since −4237≈−1.135-\frac{42}{37} \approx -1.135, the integers less than or equal to this value are −2,−3,−4,…-2, -3, -4, \dots, of which −2-2 is the greatest.

Key Concept

Solving multi-step linear inequalities with rational coefficients, applying the inequality sign-flip rule, and finding boundary integer conditions.

Alternative Method

Instead of clearing the fractions first, you can group all terms containing xx on one side and the constant terms on the other side by finding a common denominator for only the variables and only the constants. However, clearing the fractions first is generally less prone to errors.
Estimated Time:2m 0s
Question 126Question →

At a local sports club, the initiation fee is 1515 dollars less than three times the monthly membership fee, mm. If the initiation fee is 7575 dollars, what is the monthly membership fee, in dollars?

Show answer & explanation

Answer: 30

Answer

30
The statement '15 less than three times the monthly membership fee, mm' translates to the algebraic expression 3m−153m - 15. Since the initiation fee is 7575 dollars, we can set up the equation 3m−15=753m - 15 = 75. Adding 1515 to both sides yields 3m=903m = 90. Dividing both sides by 33 gives m=30m = 30. Therefore, the monthly membership fee is 3030 dollars.

Step-by-Step Solution

1
Translate the verbal description into an algebraic equation.
The expression '15 less than three times the monthly membership fee, mm' is written as 3m−153m - 15. Setting this equal to the initiation fee of 7575 gives the equation 3m−15=753m - 15 = 75.
To solve a word problem, you must first convert the written relationship into a mathematical statement.
2
Isolate the variable term by adding 1515 to both sides of the equation.
3m=903m = 90
Performing the inverse operation of subtraction (addition) simplifies the equation to isolate the term with the variable.
3
Solve for mm by dividing both sides of the equation by 33.
m=30m = 30
Performing the inverse operation of multiplication (division) isolates the variable mm completely.

Key Concept

Translating and Solving Algebraic Word Problems
Estimated Time:45s
Question 127Question →

For each quadratic equation on the left, solve for xx by factoring and match it to its correct solution set on the right.

Click a left item, then click its matching right item

Items

2x2+5x=32x^2 + 5x = 3
3x2−10x=−83x^2 - 10x = -8
x(x−4)=12x(x - 4) = 12

Matches

Show answer & explanation

Answer

The equation 2x2+5x=32x^2 + 5x = 3 matches the solution set {−3,12}\{-3, \frac{1}{2}\}; the equation 3x2−10x=−83x^2 - 10x = -8 matches the solution set {43,2}\{\frac{4}{3}, 2\}; and the equation x(x−4)=12x(x - 4) = 12 matches the solution set {−2,6}\{-2, 6\}.
Each of the quadratic equations can be solved by first rearranging the terms to set the equation equal to zero. After rewriting them in the standard form ax2+bx+c=0ax^2 + bx + c = 0, they can be factored into a product of linear binomials. Setting each factor equal to zero and solving for xx yields the solutions. Specifically: 2x2+5x=32x^2 + 5x = 3 simplifies to 2x2+5x−3=02x^2 + 5x - 3 = 0, which factors as (2x−1)(x+3)=0(2x - 1)(x + 3) = 0 and gives the solution set {−3,12}\{-3, \frac{1}{2}\}. 3x2−10x=−83x^2 - 10x = -8 simplifies to 3x2−10x+8=03x^2 - 10x + 8 = 0, which factors as (3x−4)(x−2)=0(3x - 4)(x - 2) = 0 and gives the solution set {43,2}\{\frac{4}{3}, 2\}. x(x−4)=12x(x - 4) = 12 simplifies to x2−4x−12=0x^2 - 4x - 12 = 0, which factors as (x−6)(x+2)=0(x - 6)(x + 2) = 0 and gives the solution set {−2,6}\{-2, 6\}.

Step-by-Step Solution

1
Rearrange the equation 2x2+5x=32x^2 + 5x = 3 into the standard form ax2+bx+c=0ax^2 + bx + c = 0.
2x2+5x−3=02x^2 + 5x - 3 = 0
To apply factoring and the zero product property, the quadratic expression must equal zero.
2
Factor the trinomial 2x2+5x−32x^2 + 5x - 3.
(2x−1)(x+3)=0(2x - 1)(x + 3) = 0
Since the constant term is negative and the middle coefficient is positive, the factors must have opposite signs.
3
Apply the zero product property to find the solutions for the first equation.
x=12x = \frac{1}{2} and x=−3x = -3, yielding the set {−3,12}\{-3, \frac{1}{2}\}
Setting the individual linear factors 2x−12x - 1 and x+3x + 3 to zero gives the solutions.
4
Rearrange the equation 3x2−10x=−83x^2 - 10x = -8 into standard form.
3x2−10x+8=03x^2 - 10x + 8 = 0
Add 88 to both sides to set the right side to zero.
5
Factor the trinomial 3x2−10x+83x^2 - 10x + 8.
(3x−4)(x−2)=0(3x - 4)(x - 2) = 0
The constant term is positive and the middle coefficient is negative, meaning both constant terms in the binomial factors must be negative.
6
Solve for xx using the zero product property.
x=43x = \frac{4}{3} and x=2x = 2, yielding the set {43,2}\{\frac{4}{3}, 2\}
Setting 3x−4=03x - 4 = 0 gives x=43x = \frac{4}{3}, and setting x−2=0x - 2 = 0 gives x=2x = 2.
7
Expand and rearrange the equation x(x−4)=12x(x - 4) = 12 into standard form.
x2−4x−12=0x^2 - 4x - 12 = 0
Distribute the xx on the left side to get x2−4xx^2 - 4x and subtract 1212 from both sides to set the equation to zero.
8
Factor the trinomial x2−4x−12x^2 - 4x - 12.
(x−6)(x+2)=0(x - 6)(x + 2) = 0
Find two numbers that multiply to −12-12 and add to −4-4. Those numbers are −6-6 and +2+2.
9
Solve for xx using the zero product property.
x=6x = 6 and x=−2x = -2, yielding the set {−2,6}\{-2, 6\}
Setting x−6=0x - 6 = 0 gives x=6x = 6, and setting x+2=0x + 2 = 0 gives x=−2x = -2.

Key Concept

Solving quadratic equations by rewriting them in standard form, factoring the trinomials, and using the zero product property.
Question 128Question →

When the product of the polynomials (x3−3x2+2x−4)(x^3 - 3x^2 + 2x - 4) and (ax2+bx+c)(ax^2 + bx + c) is subtracted from 2x5−5x4+5x3−18x2+4x−162x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16, the resulting polynomial is equal to 00 for all real values of xx. What is the value of 4a+2b−c4a + 2b - c?

Show answer & explanation

Answer: 6

Answer

6
The correct answer is 6. By equating the corresponding coefficients of the product (x3−3x2+2x−4)(ax2+bx+c)(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) to the polynomial 2x5−5x4+5x3−18x2+4x−162x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16, we find a=2a = 2 from the x5x^5 terms, c=4c = 4 from the constant terms, and b=1b = 1 from the x4x^4 terms. Substituting these values into 4a+2b−c4a + 2b - c yields 4(2)+2(1)−4=64(2) + 2(1) - 4 = 6.

Step-by-Step Solution

1
Relate the product of the polynomials to the given polynomial expression.
(x3−3x2+2x−4)(ax2+bx+c)=2x5−5x4+5x3−18x2+4x−16(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16
Since subtracting the product from the given polynomial results in a polynomial that is always 00, the product must be identically equal to that polynomial.
2
Equate the leading coefficients to find the value of aa.
a=2a = 2
The highest-degree term of the product is x3⋅ax2=ax5x^3 \cdot ax^2 = ax^5, which must equal the highest-degree term on the right side, 2x52x^5.
3
Equate the constant terms to find the value of cc.
−4c=−16  ⟹  c=4-4c = -16 \implies c = 4
The constant term of the product is −4⋅c=−4c-4 \cdot c = -4c, which must equal the constant term on the right side, −16-16.
4
Equate the coefficients of the x4x^4 terms to find the value of bb.
b−3a=−5  ⟹  b−6=−5  ⟹  b=1b - 3a = -5 \implies b - 6 = -5 \implies b = 1
The x4x^4 term in the expanded product comes from (x3)(bx)+(−3x2)(ax2)=(b−3a)x4(x^3)(bx) + (-3x^2)(ax^2) = (b - 3a)x^4, which must equal the x4x^4 term on the right side, −5x4-5x^4.
5
Compute the value of the requested expression 4a+2b−c4a + 2b - c.
4(2)+2(1)−4=64(2) + 2(1) - 4 = 6
Substitute the determined values a=2a = 2, b=1b = 1, and c=4c = 4 into the expression.

Key Concept

Operations on polynomials, specifically multiplication, subtraction, and equating corresponding coefficients.

Alternative Method

Alternatively, evaluate the polynomial equation at x=2x = 2. Substituting x=2x = 2 into (x3−3x2+2x−4)(ax2+bx+c)=2x5−5x4+5x3−18x2+4x−16(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16 gives (8−12+4−4)(4a+2b+c)=64−80+40−72+8−16(8 - 12 + 4 - 4)(4a + 2b + c) = 64 - 80 + 40 - 72 + 8 - 16, which simplifies to −4(4a+2b+c)=−56-4(4a + 2b + c) = -56. Dividing both sides by −4-4 yields 4a+2b+c=144a + 2b + c = 14. Since equating the constant terms gives −4c=−16  ⟹  c=4-4c = -16 \implies c = 4, we can substitute c=4c = 4 into 4a+2b+c=144a + 2b + c = 14 to get 4a+2b+4=14  ⟹  4a+2b=104a + 2b + 4 = 14 \implies 4a + 2b = 10. Subtracting c=4c=4 from both sides gives the desired expression value: 4a+2b−c=10−4=64a + 2b - c = 10 - 4 = 6.
Estimated Time:2m 30s
Question 129Question →

When the expression 2a(a2−3ab)−3b(a2+2b2)−(a3−5ab2)2a(a^2 - 3ab) - 3b(a^2 + 2b^2) - (a^3 - 5ab^2) is simplified by combining like terms, what is the coefficient of the a2ba^2b term?

Show answer & explanation

Answer: -9

Answer

The coefficient of the a2ba^2b term is −9-9.
Expanding the entire expression yields 2a3−6a2b−3a2b−6b3−a3+5ab22a^3 - 6a^2b - 3a^2b - 6b^3 - a^3 + 5ab^2. Combining the a2ba^2b terms gives (−6−3)a2b=−9a2b(-6 - 3)a^2b = -9a^2b. Therefore, the coefficient of the a2ba^2b term is −9-9.

Step-by-Step Solution

1
Distribute 2a2a to the terms inside the first set of parentheses: 2a(a2−3ab)2a(a^2 - 3ab)
2a3−6a2b2a^3 - 6a^2b
To expand the first part of the expression.
2
Distribute −3b-3b to the terms inside the second set of parentheses: −3b(a2+2b2)-3b(a^2 + 2b^2)
−3a2b−6b3-3a^2b - 6b^3
To expand the second part of the expression, ensuring the negative sign is distributed to all terms inside.
3
Distribute the negative sign to the terms inside the third set of parentheses: −(a3−5ab2)-(a^3 - 5ab^2)
−a3+5ab2-a^3 + 5ab^2
To expand the third part of the expression, reversing the sign of each term inside.
4
Identify and combine the like terms for the a2ba^2b variable combination: −6a2b−3a2b-6a^2b - 3a^2b
−9a2b-9a^2b
To simplify the expression by combining terms with the same variable components.

Key Concept

Simplifying expressions by distributing terms and combining like terms
Estimated Time:1m 30s
Question 130Question →

If aa and bb are non-zero real numbers, which of the following expressions is equivalent to a4b−2(a2b−3)2\frac{a^4 b^{-2}}{(a^2 b^{-3})^2}?

Show answer & explanation

Answer: b4b^4

Answer

b4b^4
To simplify the expression, first apply the power of a product rule to the denominator: (a2b−3)2=(a2)2(b−3)2=a4b−6(a^2 b^{-3})^2 = (a^2)^2 (b^{-3})^2 = a^4 b^{-6}. Next, divide the numerator by the simplified denominator by subtracting the exponents of the corresponding bases: a4b−2a4b−6=a4−4b−2−(−6)=a0b4\frac{a^4 b^{-2}}{a^4 b^{-6}} = a^{4-4} b^{-2 - (-6)} = a^0 b^4. Since a0=1a^0 = 1 for any non-zero real number aa, the expression simplifies to b4b^4.

Step-by-Step Solution

1
Simplify the denominator using the power of a product and power of a power rules.
(a2b−3)2=(a2)2⋅(b−3)2=a4b−6(a^2 b^{-3})^2 = (a^2)^2 \cdot (b^{-3})^2 = a^4 b^{-6}
When raising a product to a power, raise each factor to that power. When raising a power to a power, multiply the exponents.
2
Substitute the simplified denominator back into the original fraction.
a4b−2a4b−6\frac{a^4 b^{-2}}{a^4 b^{-6}}
To prepare the expression for division.
3
Divide the numerator by the denominator by subtracting the exponents of like bases.
a4−4b−2−(−6)=a0b4=1⋅b4=b4a^{4-4} b^{-2 - (-6)} = a^0 b^4 = 1 \cdot b^4 = b^4
When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Properties of exponents, specifically the power of a product rule, power of a power rule, and quotient rule.
Question 131Question →

For positive real values of uu and vv, the expression (9u−1v24u3v−4)−1/2\left( \frac{9u^{-1}v^2}{4u^3v^{-4}} \right)^{-1/2} can be simplified to which of the following?

Show answer & explanation

Answer: 2u23v3\frac{2u^2}{3v^3}

Answer

The expression is equivalent to 2u23v3\frac{2u^2}{3v^3}.
The correct answer is obtained by first simplifying the quotient inside the parenthesis to get 94u−4v6\frac{9}{4}u^{-4}v^6. Then, raising each factor to the −1/2-1/2 power yields (94)−1/2=23\left(\frac{9}{4}\right)^{-1/2} = \frac{2}{3}, (u−4)−1/2=u2(u^{-4})^{-1/2} = u^2, and (v6)−1/2=v−3(v^6)^{-1/2} = v^{-3}. Combining these results and rewriting with positive exponents gives the simplified expression.

Step-by-Step Solution

1
Simplify the uu terms inside the parenthesis using the quotient rule for exponents.
u−1u3=u−1−3=u−4\frac{u^{-1}}{u^3} = u^{-1 - 3} = u^{-4}
The quotient rule states that xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.
2
Simplify the vv terms inside the parenthesis using the quotient rule for exponents.
v2v−4=v2−(−4)=v6\frac{v^2}{v^{-4}} = v^{2 - (-4)} = v^6
Subtracting a negative exponent is equivalent to adding its absolute value.
3
Apply the outer exponent of −1/2-1/2 to the coefficient.
(94)−1/2=(49)1/2=23\left(\frac{9}{4}\right)^{-1/2} = \left(\frac{4}{9}\right)^{1/2} = \frac{2}{3}
A negative exponent represents taking the reciprocal of the base, and a fractional exponent of 1/21/2 represents the square root.
4
Apply the outer exponent of −1/2-1/2 to the simplified variable terms using the power of a power rule.
(u−4)−1/2=u2(u^{-4})^{-1/2} = u^2 and (v6)−1/2=v−3(v^6)^{-1/2} = v^{-3}
The power of a power rule states that (xa)b=xab(x^a)^b = x^{ab}.
5
Combine the simplified parts and rewrite the expression with positive exponents.
23u2v−3=2u23v3\frac{2}{3} u^2 v^{-3} = \frac{2u^2}{3v^3}
An expression with a negative exponent in the numerator can be moved to the denominator with a positive exponent.

Key Concept

Applying product, quotient, and power rules of exponents to algebraic expressions with negative and rational exponents.
Question 132Question →

For what value of kk does the equation 15(kx−3)−13(2x−5)=2\frac{1}{5}(kx - 3) - \frac{1}{3}(2x - 5) = 2 have a solution of x=7x = 7?

Show answer & explanation

Answer: 4

Answer

The value of kk is 44.
Substituting x=7x = 7 reduces the equation to 15(7k−3)−3=2\frac{1}{5}(7k - 3) - 3 = 2. Adding 3 to both sides yields 15(7k−3)=5\frac{1}{5}(7k - 3) = 5. Multiplying by 5 gives 7k−3=257k - 3 = 25. Adding 3 gives 7k=287k = 28, which results in k=4k = 4.

Step-by-Step Solution

1
Substitute x=7x = 7 into the equation.
15(7k−3)−13(2(7)−5)=2\frac{1}{5}(7k - 3) - \frac{1}{3}(2(7) - 5) = 2
Since x=7x = 7 is given as a solution, substituting it into the equation must make the equality true.
2
Evaluate and simplify the expression in the second term.
13(14−5)=13(9)=3\frac{1}{3}(14 - 5) = \frac{1}{3}(9) = 3
Follow the order of operations by simplifying the expression inside the parentheses first.
3
Isolate the fractional term containing the variable kk.
15(7k−3)−3=2  ⟹  15(7k−3)=5\frac{1}{5}(7k - 3) - 3 = 2 \implies \frac{1}{5}(7k - 3) = 5
Add 3 to both sides of the equation to eliminate the subtraction of 3.
4
Clear the fraction and solve the remaining linear equation for kk.
7k−3=25  ⟹  7k=28  ⟹  k=47k - 3 = 25 \implies 7k = 28 \implies k = 4
Multiply both sides by 5 to eliminate the denominator, add 3 to isolate the term with kk, and divide by 7.

Key Concept

Solving multi-step linear equations containing parameters and fractions.
Estimated Time:1m 30s
Question 133Question →

If m=−3m = -3, n=18n = \frac{1}{8}, and p=−2p = -2, what is the value of the algebraic expression m−2−n−1/3⋅p−3m^{-2} - n^{-1/3} \cdot p^{-3}? Express your answer as a simplified fraction.

Fill in the blanks below

The value of the expression is .
Show answer & explanation

Answer

The correct answer is 13/36.
Evaluating each term individually gives m−2=19m^{-2} = \frac{1}{9}, n−1/3=2n^{-1/3} = 2, and p−3=−18p^{-3} = -\frac{1}{8}. Substituting these values into the expression yields 19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right). Following the order of operations, we first perform the multiplication: 2⋅(−18)=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{1}{4}. We then subtract this result from the first term: 19−(−14)=19+14=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{13}{36}.

Step-by-Step Solution

1
Evaluate m−2m^{-2} when m=−3m = -3.
m−2=(−3)−2=1(−3)2=19m^{-2} = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
A negative exponent represents the reciprocal of the base raised to the positive power, and a negative base raised to an even power yields a positive result.
2
Evaluate n−1/3n^{-1/3} when n=18n = \frac{1}{8}.
n−1/3=(18)−1/3=(8−1)−1/3=81/3=2n^{-1/3} = \left(\frac{1}{8}\right)^{-1/3} = \left(8^{-1}\right)^{-1/3} = 8^{1/3} = 2
Apply the negative exponent rule to find the reciprocal of the fraction, then find the cube root of 8.
3
Evaluate p−3p^{-3} when p=−2p = -2.
p−3=(−2)−3=1(−2)3=−18p^{-3} = (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}
A negative exponent represents the reciprocal, and a negative base raised to an odd power yields a negative result.
4
Substitute the evaluated terms back into the original algebraic expression.
19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right)
Replace each variable expression with its calculated numerical value.
5
Perform the multiplication before subtraction following the order of operations.
2⋅(−18)=−28=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{2}{8} = -\frac{1}{4}
The order of operations (PEMDAS/GEMS) dictates that multiplication must be performed before subtraction.
6
Subtract the product from the first term.
19−(−14)=19+14=436+936=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}
Subtracting a negative value is equivalent to addition. Find a common denominator to add the fractions.

Key Concept

Evaluating algebraic expressions involving negative bases, negative exponents, fractional exponents, and the order of operations.
Question 134Question →

An online retailer sells two types of subscription boxes: a Basic Box and a Premium Box. Last month, the retailer sold a total of 250250 boxes. The number of Basic Boxes sold was 1010 more than 33 times the number of Premium Boxes sold. If the retailer made a total profit of 4,3504,350 dollars, and the profit from each Premium Box is 55 dollars less than twice the profit from each Basic Box, what is the profit, in dollars, for a single Premium Box?

Show answer & explanation

Answer: 25

Answer

The profit for a single Premium Box is 2525 dollars.
Solving the system of equations for the box quantities gives 190190 Basic Boxes and 6060 Premium Boxes. Setting up the profit relation p=2b−5p = 2b - 5 and the total profit equation 190b+60p=4350190b + 60p = 4350, we substitute b=p+52b = \frac{p+5}{2} to get 155p=3875155p = 3875, which simplifies to p=25p = 25 dollars.

Step-by-Step Solution

1
Define variables for the quantities of boxes sold and set up a system of equations.
Let BB be the number of Basic Boxes sold and PP be the number of Premium Boxes sold. The given relationships are: B+P=250B + P = 250 and B=3P+10B = 3P + 10.
This translates the word problem statements about the number of boxes into solvable linear equations.
2
Solve for the quantities of each box sold.
Substitute B=3P+10B = 3P + 10 into the first equation: (3P+10)+P=250  ⟹  4P+10=250  ⟹  4P=240  ⟹  P=60(3P + 10) + P = 250 \implies 4P + 10 = 250 \implies 4P = 240 \implies P = 60. Then, B=3(60)+10=190B = 3(60) + 10 = 190.
Finding the exact number of each type of box sold is required to formulate the profit equation.
3
Define variables for the profit of each box and set up the profit equations.
Let bb be the profit of a Basic Box and pp be the profit of a Premium Box. The total profit equation is 190b+60p=4350190b + 60p = 4350. The relationship between the profits is p=2b−5p = 2b - 5, which can be rearranged to b=p+52b = \frac{p+5}{2}.
This sets up the system of equations representing the profit values.
4
Substitute and solve for the profit of a Premium Box (pp).
Substitute b=p+52b = \frac{p+5}{2} into the profit equation: 190(p+52)+60p=4350  ⟹  95(p+5)+60p=4350  ⟹  95p+475+60p=4350  ⟹  155p=3875  ⟹  p=25190\left(\frac{p+5}{2}\right) + 60p = 4350 \implies 95(p+5) + 60p = 4350 \implies 95p + 475 + 60p = 4350 \implies 155p = 3875 \implies p = 25.
Solving this single-variable equation gives the final required value for the profit of a single Premium Box.

Key Concept

Translating and Solving Multi-Step Algebraic Word Problems
Question 135Question →

The algebraic expression (x2)ax−4\frac{(x^2)^a}{x^{-4}} is equivalent to x10x^{10} for all non-zero real numbers xx. What is the value of aa?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.
The correct value for aa is 33. Applying the power of a power rule to (x2)a(x^2)^a yields x2ax^{2a}. Then, applying the quotient rule to x2ax−4\frac{x^{2a}}{x^{-4}} yields x2a−(−4)=x2a+4x^{2a - (-4)} = x^{2a+4}. Equating the exponents gives 2a+4=102a + 4 = 10, which solves to a=3a = 3.

Step-by-Step Solution

1
Apply the power of a power property to the numerator.
(x2)a=x2a(x^2)^a = x^{2a}
When raising a power to another power, multiply the exponents: (xm)n=xmn(x^m)^n = x^{mn}.
2
Apply the quotient property of exponents to simplify the fraction.
x2ax−4=x2a−(−4)=x2a+4\frac{x^{2a}}{x^{-4}} = x^{2a - (-4)} = x^{2a + 4}
When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator: xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.
3
Set the resulting exponent equal to the exponent of the equivalent expression and solve for aa.
2a+4=10  ⟹  2a=6  ⟹  a=32a + 4 = 10 \implies 2a = 6 \implies a = 3
Since the bases are identical and the expressions are equivalent, their exponents must be equal.

Key Concept

Properties of Exponents in Algebraic Expressions
Question 136Question →

For a certain value of xx, the expression 25x+3\frac{2}{5}x + 3 is equal to 1111. What is the value of 2x+52x + 5?

Show answer & explanation

Answer: 45

Answer

45
To find the value of 2x+52x + 5, we first isolate the variable xx in the equation 25x+3=11\frac{2}{5}x + 3 = 11. Subtracting 33 from both sides gives 25x=8\frac{2}{5}x = 8. Multiplying both sides by the reciprocal of the coefficient, 52\frac{5}{2}, yields x=8×52=20x = 8 \times \frac{5}{2} = 20. Finally, substituting x=20x = 20 into the expression 2x+52x + 5 gives 2(20)+5=40+5=452(20) + 5 = 40 + 5 = 45.

Step-by-Step Solution

1
Set up the equation based on the text and subtract 3 from both sides of the equation to isolate the variable term.
25x=8\frac{2}{5}x = 8
To solve for xx, we first need to isolate the term containing xx.
2
Multiply both sides of the equation by 52\frac{5}{2} to solve for xx.
x=20x = 20
Multiplying by the reciprocal of the coefficient cancels the fraction, leaving xx isolated.
3
Substitute x=20x = 20 into the expression 2x+52x + 5.
2(20)+5=452(20) + 5 = 45
The question asks for the value of 2x+52x + 5, so we evaluate it using the solved value of xx.

Key Concept

Solving Linear Equations

Alternative Method

We can also multiply the entire equation by 55 first to eliminate the fraction: 2x+15=552x + 15 = 55. Subtracting 1515 from both sides gives 2x=402x = 40. Since the target expression is 2x+52x + 5, we can simply add 55 to both sides of 2x=402x = 40 to get 2x+5=452x + 5 = 45, avoiding the need to solve for xx directly.
Estimated Time:45s
Question 137Question →

An online store sells digital songs for $1.20\$1.20 each. A customer uses a discount code to get $3.00\$3.00 off the total purchase. If the customer's total cost after using the discount code is $15.00\$15.00, how many digital songs did the customer purchase?

Show answer & explanation

Answer: 15

Answer

The customer purchased 15 songs.
The word problem translates directly to the linear equation 1.20s−3.00=15.001.20s - 3.00 = 15.00, where ss is the number of songs. Adding 3.003.00 to both sides of the equation yields 1.20s=18.001.20s = 18.00. Dividing both sides by 1.201.20 yields s=15s = 15.

Step-by-Step Solution

1
Define the variable and set up the equation.
Let ss be the number of digital songs purchased. The cost of ss songs is 1.20s1.20s. Subtracting the discount of $3.00\$3.00 gives the equation: 1.20s−3.00=15.001.20s - 3.00 = 15.00.
We must represent the cost of the songs and the discount algebraically to equal the final payment.
2
Isolate the variable term.
1.20s=18.001.20s = 18.00
Adding 3.003.00 to both sides of the equation simplifies the equation by canceling the subtraction of 3.003.00.
3
Solve for the variable.
s=15s = 15
Dividing both sides of the equation by 1.201.20 isolates ss to find the total number of songs.

Key Concept

Translating real-world scenarios with linear relationships into algebraic equations and solving them.
Question 138Question →
For all positive real numbers xx and yy, the expression
(3x2y−3)3⋅(2x−1y2)2(6x3y−2)2\frac{(3x^2 y^{-3})^3 \cdot (2x^{-1} y^2)^2}{(6x^3 y^{-2})^2}
can be simplified to the form AxaybA x^a y^b, where AA, aa, and bb are integers. What is the value of the sum A+a+bA + a + b?
Show answer & explanation

Answer: 0

Answer

The value of the sum A+a+bA + a + b is 0.
By applying the rules of exponents systematically, the expression simplifies to 3x−2y−13 x^{-2} y^{-1}. Comparing this to AxaybA x^a y^b yields A=3A = 3, a=−2a = -2, and b=−1b = -1. The sum is 3+(−2)+(−1)=03 + (-2) + (-1) = 0.

Step-by-Step Solution

1
Simplify the first term in the numerator
27x6y−927x^6y^{-9}
Apply the power of a product rule and power of a power rule to (3x2y−3)3(3x^2 y^{-3})^3.
2
Simplify the second term in the numerator
4x−2y44x^{-2}y^4
Apply the power of a product rule and power of a power rule to (2x−1y2)2(2x^{-1} y^2)^2.
3
Multiply the simplified terms in the numerator together
108x4y−5108x^4y^{-5}
Multiply coefficients and add the exponents of like bases.
4
Simplify the denominator
36x6y−436x^6y^{-4}
Apply the power of a product rule and power of a power rule to (6x3y−2)2(6x^3 y^{-2})^2.
5
Divide the numerator by the denominator
3x−2y−13x^{-2}y^{-1}
Divide the coefficients and subtract the denominator exponents from the numerator exponents for like bases.
6
Sum the constants AA, aa, and bb
0
Identify A=3A = 3, a=−2a = -2, b=−1b = -1 from the expression 3x−2y−13x^{-2}y^{-1}, and calculate 3+(−2)+(−1)=03 + (-2) + (-1) = 0.

Key Concept

Properties of Exponents in Algebraic Expressions
Question 139Question →

If 2.5(x−4)=152.5(x - 4) = 15, what is the value of xx?

Show answer & explanation

Answer: 10

Answer

The value of xx is 1010.
To solve the equation 2.5(x−4)=152.5(x - 4) = 15, divide both sides by 2.52.5 to obtain x−4=6x - 4 = 6. Then, add 44 to both sides to get the final solution of x=10x = 10. Alternatively, distribute 2.52.5 to get 2.5x−10=152.5x - 10 = 15, add 1010 to both sides to get 2.5x=252.5x = 25, and divide by 2.52.5 to get x=10x = 10.

Step-by-Step Solution

1
Divide both sides of the equation by 2.52.5.
x−4=6x - 4 = 6
To isolate the parenthetical term on the left side of the equation.
2
Add 44 to both sides of the equation.
x=10x = 10
To isolate the variable xx.

Key Concept

Solving multi-step linear equations using inverse operations.

Alternative Method

Distribute 2.52.5 to get 2.5x−10=152.5x - 10 = 15. Add 1010 to both sides of the equation to get 2.5x=252.5x = 25. Divide both sides by 2.52.5 to find x=10x = 10.
Estimated Time:45s
Question 140Question →

A logistics company determines that its daily operating cost, CC (in dollars), for a delivery truck satisfies the inequality a(2C−3)3+54≤C−a2−78\frac{a(2C - 3)}{3} + \frac{5}{4} \leq \frac{C - a}{2} - \frac{7}{8}, where aa is a constant regional fuel efficiency parameter such that a<−2a < -2. Which of the following represents the range of possible operating costs CC?

Show answer & explanation

Answer: C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}

Answer

The range of possible operating costs is C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}.
To solve the inequality, we first eliminate the denominators by multiplying the entire inequality by 24, resulting in 8a(2C−3)+30≤12(C−a)−218a(2C - 3) + 30 \leq 12(C - a) - 21. Expanding both sides and gathering all terms with CC on the left gives (16a−12)C≤12a−51(16a - 12)C \leq 12a - 51. Because a<−2a < -2, the coefficient 16a−1216a - 12 is negative. Dividing by a negative number reverses the inequality direction, giving C≥12a−5116a−12C \geq \frac{12a - 51}{16a - 12}. Multiplying the numerator and denominator by −1-1 yields the correct solution.

Step-by-Step Solution

1
Clear the denominators by multiplying all terms by the least common multiple of 3, 4, 2, and 8, which is 24.
8a(2C−3)+30≤12(C−a)−218a(2C - 3) + 30 \leq 12(C - a) - 21
Eliminating fractions simplifies the algebraic manipulation of the linear inequality.
2
Expand both sides of the inequality.
16aC−24a+30≤12C−12a−2116aC - 24a + 30 \leq 12C - 12a - 21
Distributing terms allows grouping the variable CC and the constants.
3
Isolate the terms containing CC on the left side and all other terms on the right side.
16aC−12C≤12a−5116aC - 12C \leq 12a - 51
Grouping like terms is necessary to solve for CC.
4
Factor out CC on the left side.
(16a−12)C≤12a−51(16a - 12)C \leq 12a - 51
This isolates the variable CC with a single coefficient.
5
Determine the sign of the coefficient (16a−12)(16a - 12) based on the condition a<−2a < -2.
Since a<−2a < -2, we have 16a<−3216a < -32, which implies 16a−12<−4416a - 12 < -44. Thus, the coefficient is negative.
Knowing whether the coefficient is positive or negative determines whether the inequality sign must flip upon division.
6
Divide both sides by (16a−12)(16a - 12) and reverse the inequality sign.
C≥12a−5116a−12C \geq \frac{12a - 51}{16a - 12}
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.
7
Simplify the resulting fraction by multiplying the numerator and denominator by −1-1.
C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}
This yields the simplified final expression matching the target choice.

Key Concept

Solving linear inequalities involving fractions and variable parameters, with strict application of the inequality sign-flip rule when dividing by a negative algebraic term.
Estimated Time:3m 0s
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