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Elementary Algebra

302 questions

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Question 281Question →

If x=−3x = -3, y=−13y = -\frac{1}{3}, and z=16z = 16, what is the value of the algebraic expression x3y−2+3z1/2(xy+2)3−32z1/2\frac{x^3 y^{-2} + 3 z^{1/2}}{(xy + 2)^3 - \frac{3}{2} z^{1/2}}?

Show answer & explanation

Answer: -11

Answer

The evaluated value of the expression is -11.
Substituting the given values into the numerator yields (−3)3(−13)−2+3(16)1/2=(−27)(9)+3(4)=−243+12=−231(-3)^3 \left(-\frac{1}{3}\right)^{-2} + 3(16)^{1/2} = (-27)(9) + 3(4) = -243 + 12 = -231. Substituting into the denominator yields ((−3)(−13)+2)3−32(16)1/2=(1+2)3−32(4)=27−6=21\left((-3)\left(-\frac{1}{3}\right) + 2\right)^3 - \frac{3}{2}(16)^{1/2} = (1 + 2)^3 - \frac{3}{2}(4) = 27 - 6 = 21. Dividing −231-231 by 2121 produces −11-11.

Step-by-Step Solution

1
Evaluate the terms in the numerator
Numerator = -231
Calculate x3=(−3)3=−27x^3 = (-3)^3 = -27, y−2=(−13)−2=9y^{-2} = \left(-\frac{1}{3}\right)^{-2} = 9, giving x3y−2=−243x^3 y^{-2} = -243. Then calculate 3z1/2=3(4)=123 z^{1/2} = 3(4) = 12. Adding these gives −243+12=−231-243 + 12 = -231.
2
Evaluate the terms in the denominator
Denominator = 21
Calculate xy=(−3)(−13)=1xy = (-3)\left(-\frac{1}{3}\right) = 1, so (xy+2)3=(1+2)3=27(xy + 2)^3 = (1 + 2)^3 = 27. Then calculate 32z1/2=32(4)=6\frac{3}{2} z^{1/2} = \frac{3}{2}(4) = 6. Subtracting gives 27−6=2127 - 6 = 21.
3
Divide the evaluated numerator by the evaluated denominator
-11
−23121=−11\frac{-231}{21} = -11.

Key Concept

Evaluating Algebraic Expressions with Negative and Fractional Exponents
Question 282Question →

If p=−3p = -3, q=−12q = -\frac{1}{2}, and r=8r = 8, what is the value of the algebraic expression below?

p2q−3−r4/3(pq−12)2−r1/3\frac{p^2 q^{-3} - r^{4/3}}{\left(pq - \frac{1}{2}\right)^2 - r^{1/3}}

Fill in the blanks below

The value of the expression is .
Show answer & explanation

Answer

88
Evaluating each component using exponent rules and standard order of operations yields −72−16=−88-72 - 16 = -88 in the numerator and 1−2=−11 - 2 = -1 in the denominator. Dividing −88-88 by −1-1 results in the final value of 8888.

Step-by-Step Solution

1
Evaluate the terms in the numerator containing negative and rational exponents.
p2q−3=(−3)2(−12)−3=9⋅(−8)=−72p^2 q^{-3} = (-3)^2 \left(-\frac{1}{2}\right)^{-3} = 9 \cdot (-8) = -72 and r4/3=84/3=(81/3)4=24=16r^{4/3} = 8^{4/3} = (8^{1/3})^4 = 2^4 = 16.
Apply exponent rules for negative bases with integer and fractional powers: a−n=1ana^{-n} = \frac{1}{a^n} and am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m.
2
Subtract the evaluated terms to determine the total numerator value.
\text{Numerator} = -72 - 16 = -88.
Combine the evaluated terms according to the numerator expression p2q−3−r4/3p^2 q^{-3} - r^{4/3}.
3
Evaluate the grouped and exponential terms in the denominator.
pq−12=(−3)(−12)−12=32−12=1pq - \frac{1}{2} = (-3)\left(-\frac{1}{2}\right) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1. Then (pq−12)2=12=1\left(pq - \frac{1}{2}\right)^2 = 1^2 = 1, and r1/3=81/3=2r^{1/3} = 8^{1/3} = 2.
Follow the order of operations by simplifying inside the parentheses first, then applying the exponent.
4
Calculate the denominator and divide the numerator by the denominator.
\text{Denominator} = 1 - 2 = -1 .Dividingthenumeratorbythedenominatorgives. Dividing the numerator by the denominator gives \frac{-88}{-1} = 88$.
Perform the final subtraction in the denominator and divide to simplify the fraction fully.

Key Concept

Evaluating Algebraic Expressions with Negative and Fractional Exponents
Question 283Question →

If r=−3r = -3, s=2s = 2, and t=−12t = -\frac{1}{2}, what is the value of the algebraic expression r2s−t−2r+st\frac{r^2 s - t^{-2}}{r + s t}?

Show answer & explanation

Answer: −72-\frac{7}{2}

Answer

−72-\frac{7}{2}
Substituting r=−3r = -3, s=2s = 2, and t=−12t = -\frac{1}{2} into the numerator yields (−3)2(2)−(−12)−2=(9)(2)−4=18−4=14(-3)^2(2) - \left(-\frac{1}{2}\right)^{-2} = (9)(2) - 4 = 18 - 4 = 14. Substituting into the denominator yields −3+2(−12)=−3−1=−4-3 + 2\left(-\frac{1}{2}\right) = -3 - 1 = -4. Dividing numerator by denominator gives 14−4=−72\frac{14}{-4} = -\frac{7}{2}.

Step-by-Step Solution

1
Substitute the given variable values into the numerator expression r2s−t−2r^2 s - t^{-2}.
Numerator = (−3)2(2)−(−12)−2(-3)^2 (2) - \left(-\frac{1}{2}\right)^{-2}
Ensure negative values are enclosed in parentheses during substitution.
2
Evaluate the terms in the numerator following order of operations and exponent rules.
(−3)2=9(-3)^2 = 9, so 9⋅2=189 \cdot 2 = 18. Also (−12)−2=(−2)2=4\left(-\frac{1}{2}\right)^{-2} = (-2)^2 = 4. Thus, Numerator = 18−4=1418 - 4 = 14.
A negative base squared yields a positive value, and a negative exponent takes the reciprocal of the base.
3
Substitute values into the denominator expression r+str + s t and simplify.
Denominator = −3+(2)(−12)=−3+(−1)=−4-3 + (2)\left(-\frac{1}{2}\right) = -3 + (-1) = -4
Multiply ss and tt first before adding to rr according to PEMDAS.
4
Divide the evaluated numerator by the evaluated denominator.
14−4=−72\frac{14}{-4} = -\frac{7}{2}
Simplify the fraction by dividing both numerator and denominator by 2.

Key Concept

Evaluating Algebraic Expressions with Negative Bases and Negative Exponents
Estimated Time:1m 15s
Question 284Question →
If m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4, what is the value of the algebraic expression below?
m3n−2−−p⋅m2m2−n−1+p\frac{m^3 n^{-2} - \sqrt{-p \cdot m^2}}{m^2 - n^{-1} + p}
Show answer & explanation

Answer: 763\frac{76}{3}

Answer

763\frac{76}{3}
Substituting m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4 into the numerator gives (−2)3(13)−2−−(−4)(−2)2=−8(9)−16=−72−4=−76(-2)^3 \left(\frac{1}{3}\right)^{-2} - \sqrt{-(-4)(-2)^2} = -8(9) - \sqrt{16} = -72 - 4 = -76. Substituting into the denominator gives (−2)2−(13)−1+(−4)=4−3−4=−3(-2)^2 - \left(\frac{1}{3}\right)^{-1} + (-4) = 4 - 3 - 4 = -3. Dividing −76-76 by −3-3 gives 763\frac{76}{3}.

Step-by-Step Solution

1
Evaluate the terms in the numerator
m3=(−2)3=−8m^3 = (-2)^3 = -8, n−2=(13)−2=9n^{-2} = \left(\frac{1}{3}\right)^{-2} = 9, and −p⋅m2=−(−4)⋅(−2)2=4⋅4=4\sqrt{-p \cdot m^2} = \sqrt{-(-4) \cdot (-2)^2} = \sqrt{4 \cdot 4} = 4. Thus, Numerator =(−8)(9)−4=−72−4=−76= (-8)(9) - 4 = -72 - 4 = -76.
Simplify each term in the numerator using proper exponent rules and sign conventions.
2
Evaluate the terms in the denominator
m2=(−2)2=4m^2 = (-2)^2 = 4, n−1=(13)−1=3n^{-1} = \left(\frac{1}{3}\right)^{-1} = 3, and p=−4p = -4. Thus, Denominator =4−3+(−4)=1−4=−3= 4 - 3 + (-4) = 1 - 4 = -3.
Substitute the variable values into the denominator expression and evaluate left to right.
3
Divide the numerator by the denominator
−76−3=763.\frac{-76}{-3} = \frac{76}{3}.
Dividing two negative numbers yields a positive quotient.

Key Concept

Evaluating algebraic expressions with negative bases, fractional exponents, and order of operations
Estimated Time:2m 0s
Question 285Question →
If p=−3p = -3, q=12q = \frac{1}{2}, and r=−27r = -27, what is the value of the algebraic expression below?
p2q−3+r1/3(2pq+5)2+q−1\frac{p^2 q^{-3} + r^{1/3}}{(2pq + 5)^2 + q^{-1}}
Show answer & explanation

Answer: 11.5

Answer

11.5
Evaluating each component of the expression step-by-step using order of operations:
1. Numerator: p2q−3+r1/3=(−3)2(12)−3+(−27)1/3=9⋅8+(−3)=72−3=69p^2 q^{-3} + r^{1/3} = (-3)^2 \left(\frac{1}{2}\right)^{-3} + (-27)^{1/3} = 9 \cdot 8 + (-3) = 72 - 3 = 69.
2. Denominator: (2pq+5)2+q−1=(2(−3)(12)+5)2+(12)−1=(−3+5)2+2=22+2=4+2=6(2pq + 5)^2 + q^{-1} = \left(2(-3)\left(\frac{1}{2}\right) + 5\right)^2 + \left(\frac{1}{2}\right)^{-1} = (-3 + 5)^2 + 2 = 2^2 + 2 = 4 + 2 = 6.
3. Final Quotient: 696=11.5\frac{69}{6} = 11.5.

Step-by-Step Solution

1
Evaluate the terms in the numerator
Numerator = 69
Since p=−3p = -3, p2=9p^2 = 9. With q=12q = \frac{1}{2}, q−3=23=8q^{-3} = 2^3 = 8. Thus, p2q−3=9⋅8=72p^2 q^{-3} = 9 \cdot 8 = 72. Also, r1/3=(−27)1/3=−3r^{1/3} = (-27)^{1/3} = -3. Adding these values gives 72+(−3)=6972 + (-3) = 69.
2
Evaluate the terms in the denominator
Denominator = 6
First, 2pq=2(−3)(12)=−32pq = 2(-3)\left(\frac{1}{2}\right) = -3. Then (2pq+5)2=(−3+5)2=22=4(2pq + 5)^2 = (-3 + 5)^2 = 2^2 = 4. Next, q−1=(12)−1=2q^{-1} = \left(\frac{1}{2}\right)^{-1} = 2. Adding these components gives 4+2=64 + 2 = 6.
3
Divide the numerator by the denominator
11.5
Dividing the numerator (69) by the denominator (6) yields 696=11.5\frac{69}{6} = 11.5 (or 232\frac{23}{2}).

Key Concept

Evaluating algebraic expressions with negative bases, negative exponents, fractional exponents, and order of operations.
Question 286Question →

An engineering formula used to calculate a structural load index is given by L=a3b−2−c3/4a2+12bL = \frac{a^3 b^{-2} - c^{3/4}}{a^2 + 12b}. What is the value of LL when a=−2a = -2, b=13b = \frac{1}{3}, and c=16c = 16?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.
Substituting the values into the formula gives L=(−2)3⋅(1/3)−2−163/4(−2)2+12(1/3)=−8⋅9−84+4=−72−88=−808=−10L = \frac{(-2)^3 \cdot (1/3)^{-2} - 16^{3/4}}{(-2)^2 + 12(1/3)} = \frac{-8 \cdot 9 - 8}{4 + 4} = \frac{-72 - 8}{8} = \frac{-80}{8} = -10.

Step-by-Step Solution

1
Evaluate the terms in the numerator containing powers and negative exponents
a3=−8a^3 = -8, b−2=9b^{-2} = 9, and c3/4=8c^{3/4} = 8
Negative bases raised to odd powers remain negative: (−2)3=−8(-2)^3 = -8. A negative exponent represents the reciprocal raised to a positive exponent: (1/3)−2=32=9(1/3)^{-2} = 3^2 = 9. Fractional exponent c3/4=(164)3=23=8c^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8.
2
Compute the full numerator
−72−8=−80-72 - 8 = -80
Multiply a3a^3 and b−2b^{-2} to get (−8)(9)=−72(-8)(9) = -72, then subtract c3/4=8c^{3/4} = 8.
3
Evaluate the denominator
(−2)2+12(13)=4+4=8(-2)^2 + 12\left(\frac{1}{3}\right) = 4 + 4 = 8
Squaring a negative number yields a positive value: (−2)2=4(-2)^2 = 4. Multiplying 12⋅13=412 \cdot \frac{1}{3} = 4.
4
Divide the numerator by the denominator
−808=−10\frac{-80}{8} = -10
Dividing a negative integer by a positive integer yields a negative result.

Key Concept

Evaluating algebraic expressions involving negative numbers, negative exponents, and rational exponents
Estimated Time:1m 15s
Question 287Question →
An algebraic function F(a,b)F(a, b) is defined as:
F(a,b)=a2−3bab−1+2F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2}

The values for variables aa and bb are given in the table below:

VariableValue
aa−4-4
bb−12-\frac{1}{2}

What is the value of F(a,b)F(a, b) for these given values?

Show answer & explanation

Answer: 74\frac{7}{4}

Answer

The value of F(a,b)F(a, b) is 74\frac{7}{4}.
Substituting a=−4a = -4 and b=−1/2b = -1/2 into the expression gives a numerator of (−4)2−3(−1/2)=16+3/2=35/2(-4)^2 - 3(-1/2) = 16 + 3/2 = 35/2 and a denominator of (−4)(−2)+2=8+2=10(-4)(-2) + 2 = 8 + 2 = 10. Dividing 35/235/2 by 1010 simplifies to 35/2035/20, which equals 7/47/4.

Step-by-Step Solution

1
Evaluate the reciprocal term b−1b^{-1}
b−1=(−12)−1=−2b^{-1} = \left(-\frac{1}{2}\right)^{-1} = -2
A negative exponent indicates taking the reciprocal of the base.
2
Evaluate the numerator a2−3ba^2 - 3b
(-4)^2 - 3\left(-\frac{1}{2}\right) = 16 + \frac{3}{2} = \frac{35}{2}
Squaring a negative number yields a positive value ((−4)2=16(-4)^2 = 16), and subtracting a negative value is equivalent to addition.
3
Evaluate the denominator ab−1+2a b^{-1} + 2
(-4)(-2) + 2 = 8 + 2 = 10
Multiplying two negative numbers yields a positive product.
4
Divide the numerator by the denominator and simplify
35210=3520=74\frac{\frac{35}{2}}{10} = \frac{35}{20} = \frac{7}{4}
Dividing 352\frac{35}{2} by 1010 gives 3520\frac{35}{20}, which simplifies to 74\frac{7}{4} when dividing both numerator and denominator by 55.

Key Concept

Evaluating algebraic expressions with negative numbers, fractions, and negative exponents.
Estimated Time:1m 30s
Question 288Question →

If u=−3u = -3 and v=−12v = -\frac{1}{2}, what is the value of the algebraic expression below? Enter your numerical answer in the blank.

Fill in the blanks below

The value of the expression 4uv2−u22v−u\frac{4uv^2 - u^2}{2v - u} is .
Show answer & explanation

Answer

The value of the expression when evaluated at u=−3u = -3 and v=−12v = -\frac{1}{2} is −6-6.
Substituting u=−3u = -3 and v=−12v = -\frac{1}{2} into the numerator gives 4(−3)(−12)2−(−3)2=−3−9=−124(-3)(-\frac{1}{2})^2 - (-3)^2 = -3 - 9 = -12. Substituting into the denominator gives 2(−12)−(−3)=−1+3=22(-\frac{1}{2}) - (-3) = -1 + 3 = 2. Dividing −12-12 by 22 results in −6-6.

Step-by-Step Solution

1
Evaluate the terms containing exponents in the numerator.
v2=(−12)2=14v^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} and u2=(−3)2=9u^2 = (-3)^2 = 9.
Exponents must be calculated before multiplication according to the order of operations.
2
Evaluate the numerator 4uv2−u24uv^2 - u^2.
4(−3)(14)−9=−3−9=−124(-3)\left(\frac{1}{4}\right) - 9 = -3 - 9 = -12.
Multiply 4×(−3)×144 \times (-3) \times \frac{1}{4} to get −3-3, then subtract 99.
3
Evaluate the denominator 2v−u2v - u.
2(−12)−(−3)=−1+3=22\left(-\frac{1}{2}\right) - (-3) = -1 + 3 = 2.
Multiplying 22 by −12-\frac{1}{2} gives −1-1, and subtracting negative 33 is equivalent to adding 33.
4
Divide the numerator by the denominator.
−122=−6\frac{-12}{2} = -6.
Dividing −12-12 by 22 yields the final result −6-6.

Key Concept

Evaluating algebraic expressions involving negative numbers, fractions, and order of operations.
Estimated Time:1m 30s
Question 289Question →

If a=−3a = -3 and b=−2b = -2, what is the value of the expression 2a2−b3(a−b)2+ab\frac{2a^2 - b^3}{(a - b)^2 + ab}?

Show answer & explanation

Answer: 267\frac{26}{7}

Answer

267\frac{26}{7}
Substituting a=−3a = -3 and b=−2b = -2 directly into the numerator yields 2(−3)2−(−2)3=2(9)−(−8)=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 26. Substituting into the denominator gives ((−3)−(−2))2+(−3)(−2)=(−1)2+6=7((-3) - (-2))^2 + (-3)(-2) = (-1)^2 + 6 = 7. Combining these results gives 267\frac{26}{7}.

Step-by-Step Solution

1
Substitute a=−3a = -3 and b=−2b = -2 into the numerator 2a2−b32a^2 - b^3.
2(−3)2−(−2)3=2(9)−(−8)=18+8=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 18 + 8 = 26
Squaring −3-3 gives 99, and cubing −2-2 gives −8-8. Subtracting −8-8 is equivalent to adding 88.
2
Substitute a=−3a = -3 and b=−2b = -2 into the denominator (a−b)2+ab(a - b)^2 + ab.
((−3)−(−2))2+(−3)(−2)=(−3+2)2+6=(−1)2+6=1+6=7((-3) - (-2))^2 + (-3)(-2) = (-3 + 2)^2 + 6 = (-1)^2 + 6 = 1 + 6 = 7
Subtracting a negative number becomes addition, and squaring −1-1 yields 11.
3
Form the fraction by dividing the numerator by the denominator.
267\frac{26}{7}
The evaluated numerator is 2626 and the evaluated denominator is 77.

Key Concept

Order of operations and handling negative signs when evaluating algebraic expressions involving powers and parentheses.
Estimated Time:1m 0s
Question 290Question →

If x=−3x = -3, y=−12y = -\frac{1}{2}, and z=8z = 8, what is the value of the algebraic expression x2y−1+z2/32xy−1\frac{x^2 y^{-1} + z^{2/3}}{2xy - 1}?

Show answer & explanation

Answer: -7

Answer

The value of the expression is -7.
Substituting the given values into the expression requires careful application of exponent rules and order of operations. First, (−3)2=9(-3)^2 = 9 and (−12)−1=−2(-\frac{1}{2})^{-1} = -2, so x2y−1=9×(−2)=−18x^2 y^{-1} = 9 \times (-2) = -18. Second, 82/3=(83)2=22=48^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4. This makes the numerator −18+4=−14-18 + 4 = -14. The denominator evaluates to 2(−3)(−12)−1=3−1=22(-3)(-\frac{1}{2}) - 1 = 3 - 1 = 2. Dividing −14-14 by 22 gives the correct answer of −7-7.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−12y = -\frac{1}{2} into x2y−1x^2 y^{-1}
(−3)2⋅(−2)=−18(-3)^2 \cdot (-2) = -18
Squaring −3-3 yields 99, and taking the reciprocal of −12-\frac{1}{2} yields −2-2.
2
Substitute z=8z = 8 into z2/3z^{2/3}
82/3=48^{2/3} = 4
Taking the cube root of 88 gives 22, and squaring 22 gives 44.
3
Evaluate the numerator
−18+4=−14-18 + 4 = -14
Adding the evaluated terms together.
4
Substitute values into the denominator 2xy−12xy - 1
2(−3)(−12)−1=3−1=22(-3)\left(-\frac{1}{2}\right) - 1 = 3 - 1 = 2
Multiplying 22, −3-3, and −12-\frac{1}{2} produces 33, then subtracting 11 yields 22.
5
Divide the numerator by the denominator
−142=−7\frac{-14}{2} = -7
Simplifying the rational expression yields the final numeric answer.

Key Concept

Evaluating algebraic expressions with negative bases, rational exponents, and negative exponents
Question 291Question →

If x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3, what is the value of the algebraic expression 3x2y−z2x+yz\frac{3x^2 y - z^2}{x + yz}?

Show answer & explanation

Answer: 53\frac{5}{3}

Answer

53\frac{5}{3}
Substituting the given values into the expression requires evaluating powers of negative numbers and performing multiplication prior to addition/subtraction. In the numerator, 3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2(\frac{1}{3}) - (-3)^2 = 3(4)(\frac{1}{3}) - 9 = 4 - 9 = -5. In the denominator, −2+(13)(−3)=−2−1=−3-2 + (\frac{1}{3})(-3) = -2 - 1 = -3. Simplifying the fraction −5−3\frac{-5}{-3} gives 53\frac{5}{3}.

Step-by-Step Solution

1
Evaluate the numerator 3x2y−z23x^2 y - z^2 by substituting x=−2x = -2, y=13y = \frac{1}{3}, and z=−3z = -3.
3(−2)2(13)−(−3)2=3(4)(13)−9=4−9=−53(-2)^2\left(\frac{1}{3}\right) - (-3)^2 = 3(4)\left(\frac{1}{3}\right) - 9 = 4 - 9 = -5
Squaring negative numbers yields positive values: (−2)2=4(-2)^2 = 4 and (−3)2=9(-3)^2 = 9.
2
Evaluate the denominator x+yzx + yz using the same variable values.
−2+(13)(−3)=−2+(−1)=−3-2 + \left(\frac{1}{3}\right)(-3) = -2 + (-1) = -3
Perform the multiplication y⋅zy \cdot z before adding to xx according to the order of operations.
3
Divide the evaluated numerator by the evaluated denominator.
−5−3=53\frac{-5}{-3} = \frac{5}{3}
Dividing two negative numbers produces a positive quotient.

Key Concept

Evaluating Algebraic Expressions with Negative Values and Rational Numbers
Question 292Question →

If m=−4m = -4 and n=−12n = -\frac{1}{2}, what is the value of the algebraic expression m2−4n3(m+2n)2\frac{m^2 - 4n^3}{(m + 2n)^2}?

Show answer & explanation

Answer: 3350\frac{33}{50}

Answer

3350\frac{33}{50}
Substituting m=−4m = -4 and n=−12n = -\frac{1}{2} into the numerator gives (−4)2−4(−12)3=16−(−12)=332(-4)^2 - 4\left(-\frac{1}{2}\right)^3 = 16 - \left(-\frac{1}{2}\right) = \frac{33}{2}. Substituting into the denominator gives (−4+2(−12))2=(−5)2=25\left(-4 + 2\left(-\frac{1}{2}\right)\right)^2 = (-5)^2 = 25. Dividing the numerator by the denominator yields 33/225=3350\frac{33/2}{25} = \frac{33}{50}.

Step-by-Step Solution

1
Evaluate the terms in the numerator: m2m^2 and 4n34n^3.
m2=(−4)2=16m^2 = (-4)^2 = 16 and 4n3=4(−12)3=4(−18)=−124n^3 = 4\left(-\frac{1}{2}\right)^3 = 4\left(-\frac{1}{8}\right) = -\frac{1}{2}.
Substitute m=−4m = -4 and n=−12n = -\frac{1}{2} while applying powers before multiplication according to PEMDAS.
2
Calculate the complete numerator by subtracting the evaluated terms.
m2−4n3=16−(−12)=16+12=332m^2 - 4n^3 = 16 - \left(-\frac{1}{2}\right) = 16 + \frac{1}{2} = \frac{33}{2}.
Subtracting a negative quantity is equivalent to adding its positive counterpart.
3
Evaluate the expression inside the denominator parentheses, then square it.
m+2n=−4+2(−12)=−4−1=−5m + 2n = -4 + 2\left(-\frac{1}{2}\right) = -4 - 1 = -5, and (−5)2=25(-5)^2 = 25.
Grouped operations inside parentheses must be evaluated prior to applying outer exponents.
4
Divide the numerator by the denominator to get the final simplified fraction.
33225=332×25=3350\frac{\frac{33}{2}}{25} = \frac{33}{2 \times 25} = \frac{33}{50}.
Dividing a fraction by an integer combines the denominators.

Key Concept

Evaluating algebraic expressions involving negative bases, fractions, and order of operations
Estimated Time:1m 30s
Question 293Question →

Evaluate the algebraic expression 3x2−2xy+y23x^2 - 2xy + y^2 for x=−3x = -3 and y=−2y = -2.

Fill in the blanks below

The value of the expression 3x2−2xy+y23x^2 - 2xy + y^2 when x=−3x = -3 and y=−2y = -2 is .
Show answer & explanation

Answer

19
Substituting x=−3x = -3 and y=−2y = -2 into 3x2−2xy+y23x^2 - 2xy + y^2 yields 3(−3)2−2(−3)(−2)+(−2)2=3(9)−12+4=27−12+4=193(-3)^2 - 2(-3)(-2) + (-2)^2 = 3(9) - 12 + 4 = 27 - 12 + 4 = 19.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−2y = -2 into the expression
3(−3)2−2(−3)(−2)+(−2)23(-3)^2 - 2(-3)(-2) + (-2)^2
Replace each variable with its assigned value, using parentheses to properly preserve negative signs.
2
Evaluate the exponent terms
3(9)−2(−3)(−2)+43(9) - 2(-3)(-2) + 4
Follow the order of operations (PEMDAS) by evaluating powers first: (−3)2=9(-3)^2 = 9 and (−2)2=4(-2)^2 = 4.
3
Perform the multiplications
27−12+427 - 12 + 4
Multiply the numerical factors: 3×9=273 \times 9 = 27 and −2×(−3)×(−2)=−12-2 \times (-3) \times (-2) = -12.
4
Perform addition and subtraction from left to right
1919
Subtract 1212 from 2727 to get 1515, then add 44 to arrive at 1919.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Question 294Question →

If x=−3x = -3, y=−12y = -\frac{1}{2}, and z=4z = 4, what is the numerical value of the algebraic expression x2−4y2x+yz\frac{x^2 - 4y^2}{x + yz}?

Show answer & explanation

Answer: -1.6

Answer

The numerical value of the expression is −1.6-1.6.
Substituting x=−3x = -3, y=−12y = -\frac{1}{2}, and z=4z = 4 into x2−4y2x+yz\frac{x^2 - 4y^2}{x + yz} gives a numerator of (−3)2−4(−12)2=9−4(14)=8(-3)^2 - 4\left(-\frac{1}{2}\right)^2 = 9 - 4\left(\frac{1}{4}\right) = 8 and a denominator of −3+(−12)(4)=−3−2=−5-3 + \left(-\frac{1}{2}\right)(4) = -3 - 2 = -5. Evaluating 8−5\frac{8}{-5} yields −1.6-1.6.

Step-by-Step Solution

1
Evaluate the numerator x2−4y2x^2 - 4y^2
8
Squaring −3-3 gives 99, and squaring −12-\frac{1}{2} gives 14\frac{1}{4}. Thus, 9−4(14)=9−1=89 - 4\left(\frac{1}{4}\right) = 9 - 1 = 8.
2
Evaluate the denominator x+yzx + yz
-5
Multiplying −12-\frac{1}{2} by 44 gives −2-2. Adding −3+(−2)-3 + (-2) yields −5-5.
3
Compute the final fraction quotient
-1.6
Dividing the numerator 88 by the denominator −5-5 yields −1.6-1.6.

Key Concept

Evaluating algebraic expressions using order of operations with negative bases and fractional values.
Estimated Time:1m 30s
Question 295Question →

If a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3, what is the value of the algebraic expression a3b−2c2a−b−2\frac{a^3 b - 2c^2}{a - b^{-2}}?

Show answer & explanation

Answer: 73\frac{7}{3}

Answer

The correct answer is 73\frac{7}{3}.
Substituting a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3 gives a numerator of (−2)3(−12)−2(3)2=(−8)(−12)−18=4−18=−14(-2)^3 \left(-\frac{1}{2}\right) - 2(3)^2 = (-8)\left(-\frac{1}{2}\right) - 18 = 4 - 18 = -14, and a denominator of −2−(−12)−2=−2−4=−6-2 - \left(-\frac{1}{2}\right)^{-2} = -2 - 4 = -6. Dividing −14-14 by −6-6 simplifies to 73\frac{7}{3}.

Step-by-Step Solution

1
Substitute a=−2a = -2, b=−12b = -\frac{1}{2}, and c=3c = 3 into the numerator a3b−2c2a^3 b - 2c^2.
Numerator =(−2)3⋅(−12)−2(3)2=(−8)⋅(−12)−2(9)=4−18=−14= (-2)^3 \cdot \left(-\frac{1}{2}\right) - 2(3)^2 = (-8) \cdot \left(-\frac{1}{2}\right) - 2(9) = 4 - 18 = -14.
Apply exponents first, then multiplication, and finally subtraction.
2
Substitute a=−2a = -2 and b=−12b = -\frac{1}{2} into the denominator a−b−2a - b^{-2}.
Denominator =−2−(−12)−2=−2−(−2)2=−2−4=−6= -2 - \left(-\frac{1}{2}\right)^{-2} = -2 - (-2)^2 = -2 - 4 = -6.
A negative exponent indicates the reciprocal of the base, so (−12)−2=(−2)2=4\left(-\frac{1}{2}\right)^{-2} = (-2)^2 = 4.
3
Divide the numerator by the denominator and simplify the fraction.
−14−6=146=73.\frac{-14}{-6} = \frac{14}{6} = \frac{7}{3}.
Dividing two negative numbers yields a positive quotient, which simplifies by dividing the numerator and denominator by 2.

Key Concept

Evaluating algebraic expressions with negative bases and negative integer exponents
Estimated Time:1m 30s
Question 296Question →

The polynomial 4x2−8x−54x^2 - 8x - 5 is subtracted from the polynomial 7x2−3x+47x^2 - 3x + 4. The simplified difference can be expressed as ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constant integers. What is the value of the coefficient bb?

Show answer & explanation

Answer: 5

Answer

The coefficient of the xx term, bb, is 5.
Subtracting 4x2−8x−54x^2 - 8x - 5 from 7x2−3x+47x^2 - 3x + 4 yields (7x2−4x2)+(−3x−(−8x))+(4−(−5))=3x2+5x+9(7x^2 - 4x^2) + (-3x - (-8x)) + (4 - (-5)) = 3x^2 + 5x + 9. The coefficient of the linear term xx is 55.

Step-by-Step Solution

1
Set up the subtraction expression.
(7x2−3x+4)−(4x2−8x−5)(7x^2 - 3x + 4) - (4x^2 - 8x - 5)
Subtracting the second polynomial from the first requires enclosing the second polynomial in parentheses to apply the subtraction to all terms.
2
Distribute the negative sign to all terms inside the parentheses.
7x2−3x+4−4x2+8x+57x^2 - 3x + 4 - 4x^2 + 8x + 5
Distributing the subtraction sign flips the sign of each term: positive terms become negative, and negative terms become positive.
3
Group and combine like terms.
3x2+5x+93x^2 + 5x + 9
Combine the coefficients of matching variable parts: (7−4)x2=3x2(7 - 4)x^2 = 3x^2, (−3+8)x=5x(-3 + 8)x = 5x, and 4−(−5)=94 - (-5) = 9.
4
Identify the coefficient bb corresponding to the xx term.
b=5b = 5
Comparing the simplified expression 3x2+5x+93x^2 + 5x + 9 to ax2+bx+cax^2 + bx + c shows that the coefficient of xx is 5.

Key Concept

Polynomial Subtraction and Combining Like Terms
Question 297Question →

A business owner registers 44 identical cell phone lines under a group plan. The service provider charges a monthly base fee of $15.00\$15.00 per line, plus $0.05\$0.05 per minute of call time. If the total monthly bill for all 44 lines was $96.00\$96.00 and each line used the exact same number of minutes, how many minutes did each line use?

Show answer & explanation

Answer: 180180

Answer

Each cell phone line used 180180 minutes.
The correct answer is 180180 minutes. The total bill for the 44 lines is represented by 4(15.00+0.05m)=96.004(15.00 + 0.05m) = 96.00. Dividing both sides by 44 gives the monthly charge per line: 15.00+0.05m=24.0015.00 + 0.05m = 24.00. Subtracting the base fee of 15.0015.00 from both sides leaves the total per-minute cost of 0.05m=9.000.05m = 9.00. Dividing by the per-minute rate of 0.050.05 gives m=180m = 180.

Step-by-Step Solution

1
Set up the linear equation representing the total monthly bill.
4(15.00+0.05m)=96.004(15.00 + 0.05m) = 96.00, where mm represents the number of minutes used per line.
The total bill is the sum of the charges for all 44 lines, where each line costs $15.00\$15.00 plus $0.05\$0.05 per minute.
2
Divide both sides of the equation by 44 to isolate the single-line cost expression.
15.00+0.05m=24.0015.00 + 0.05m = 24.00
Dividing both sides by 44 simplifies the equation and isolates the cost per line.
3
Subtract the base fee of 15.0015.00 from both sides of the equation.
0.05m=9.000.05m = 9.00
This isolates the variable charge term on the left side of the equation.
4
Divide both sides by 0.050.05 to solve for mm.
m=180m = 180
Dividing the remaining total of 9.009.00 by the rate of 0.050.05 per minute yields the total number of minutes used.

Key Concept

Solving Linear Equations

Alternative Method

Instead of dividing by 44 first, distribute the 44 to both terms inside the parentheses: 4(15.00)+4(0.05m)=96.004(15.00) + 4(0.05m) = 96.00. This simplifies to 60.00+0.20m=96.0060.00 + 0.20m = 96.00. Subtract 60.0060.00 from both sides to get 0.20m=36.000.20m = 36.00. Finally, divide by 0.200.20 to find m=180m = 180.
Estimated Time:1m 15s
Question 298Question →

An industrial designer is drafting a template for a rectangular solar panel. The total area of the panel is represented by the polynomial 4x3(3x2−2x+5)4x^3(3x^2 - 2x + 5) square centimeters. A rectangular sensor cutout with an area of (2x2−3)2(2x^2 - 3)^2 square centimeters is removed from the panel. Which of the following polynomials represents the remaining area, in square centimeters, of the solar panel in terms of xx?

Show answer & explanation

Answer: 12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9

Answer

12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9
To find the remaining area, calculate the total area and subtract the cutout area. The total area is 4x3(3x2−2x+5)=12x5−8x4+20x34x^3(3x^2 - 2x + 5) = 12x^5 - 8x^4 + 20x^3. The area of the cutout is (2x2−3)2=4x4−12x2+9(2x^2 - 3)^2 = 4x^4 - 12x^2 + 9. Subtracting the cutout polynomial requires distributing the negative sign to all three terms: (12x5−8x4+20x3)−(4x4−12x2+9)=12x5−8x4+20x3−4x4+12x2−9(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 8x^4 + 20x^3 - 4x^4 + 12x^2 - 9. Combining like terms yields the expression 12x5−12x4+20x3+12x2−912x^5 - 12x^4 + 20x^3 + 12x^2 - 9.

Step-by-Step Solution

1
Find the polynomial representing the total area of the solar panel by distributing 4x34x^3 through the expression (3x2−2x+5)(3x^2 - 2x + 5).
12x5−8x4+20x312x^5 - 8x^4 + 20x^3
When multiplying terms with the same base, add their exponents (xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}).
2
Find the polynomial representing the area of the sensor cutout by expanding (2x2−3)2(2x^2 - 3)^2.
4x4−12x2+94x^4 - 12x^2 + 9
Use the binomial squaring identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, where a=2x2a = 2x^2 and b=3b = 3.
3
Subtract the cutout area from the total area, ensuring the negative sign is distributed to every term in the cutout polynomial.
(12x5−8x4+20x3)−(4x4−12x2+9)=12x5−12x4+20x3+12x2−9(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 12x^4 + 20x^3 + 12x^2 - 9
Distributing the negative sign changes the signs of all terms in the subtracted polynomial, allowing like terms to be combined.

Key Concept

Polynomial subtraction and expansion using binomial squaring and exponent properties
Estimated Time:1m 30s
Question 299Question →

A chemist uses the algebraic formula E=2x2−yzx+y2E = \frac{2x^2 - yz}{x + y^2} to determine the stability index of a synthesized compound. If x=−4x = -4, y=−3y = -3, and z=23z = \frac{2}{3}, what is the value of EE?

Show answer & explanation

Answer: 345\frac{34}{5}

Answer

345\frac{34}{5}
Substituting x=−4x = -4, y=−3y = -3, and z=23z = \frac{2}{3} into the numerator yields 2(−4)2−(−3)(23)=2(16)−(−2)=32+2=342(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34. Substituting into the denominator yields −4+(−3)2=−4+9=5-4 + (-3)^2 = -4 + 9 = 5. Thus, the value of the expression is 345\frac{34}{5}.

Step-by-Step Solution

1
Substitute the given values into the numerator 2x2−yz2x^2 - yz.
2(−4)2−(−3)(23)=2(16)−(−2)=32+2=342(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34
Squaring a negative number yields a positive result (−4)2=16(-4)^2 = 16, and multiplying −3-3 by 23\frac{2}{3} gives −2-2, which is then subtracted.
2
Substitute the given values into the denominator x+y2x + y^2.
−4+(−3)2=−4+9=5-4 + (-3)^2 = -4 + 9 = 5
Evaluating the exponent first gives (−3)2=9(-3)^2 = 9, then adding −4-4 gives 55.
3
Divide the numerator by the denominator.
345\frac{34}{5}
Combine the evaluated numerator and denominator to get the final value of the expression.

Key Concept

Evaluating algebraic expressions involving multiple variables with negative bases and fractional terms using standard order of operations.
Estimated Time:1m 15s
Question 300Question →

If a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2}, what is the value of the algebraic expression a3b+4ca2−2b\frac{a^3 b + 4c}{a^2 - 2b}?

Fill in the blanks below

If a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2}, the value of the algebraic expression a3b+4ca2−2b\frac{a^3 b + 4c}{a^2 - 2b} is .
Show answer & explanation

Answer

13
Substituting a=−2a = -2, b=3b = 3, and c=−12c = -\frac{1}{2} into the given expression yields a numerator of (−2)3(3)+4(−12)=−24−2=−26(-2)^3(3) + 4(-\frac{1}{2}) = -24 - 2 = -26 and a denominator of (−2)2−2(3)=4−6=−2(-2)^2 - 2(3) = 4 - 6 = -2. Dividing the numerator by the denominator gives −26−2=13\frac{-26}{-2} = 13.

Step-by-Step Solution

1
Evaluate the terms in the numerator individually.
a3b=(−2)3⋅3=−8⋅3=−24a^3 b = (-2)^3 \cdot 3 = -8 \cdot 3 = -24 and 4c=4(−12)=−24c = 4\left(-\frac{1}{2}\right) = -2.
Exponents take precedence before multiplication, and multiplying a positive by a negative yields a negative number.
2
Combine the terms to calculate the numerator.
Numerator =−24+(−2)=−26= -24 + (-2) = -26.
Adding two negative numbers sums their magnitudes with a negative sign.
3
Evaluate the terms in the denominator.
a2=(−2)2=4a^2 = (-2)^2 = 4 and 2b=2(3)=62b = 2(3) = 6.
Squaring a negative base results in a positive value.
4
Calculate the denominator.
Denominator =4−6=−2= 4 - 6 = -2.
Subtracting a larger number from a smaller number produces a negative result.
5
Divide the numerator by the denominator to find the final value.
−26−2=13\frac{-26}{-2} = 13.
Dividing a negative number by a negative number yields a positive quotient.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 30s
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