Equivalent Algebraic Expressions Practice Questions — SAT

Question 1Question →

For all $x > 1$ , which of the following is equivalent to the expression $\frac{2x^2 + 3x - 3}{x - 1} - \frac{x^3 + 2x^2 + x}{x^2 - 1}$ ?

x - 3

x + 3

x + 4

x + 5

Show answer & explanation

Answer: $x + 3$

Answer

x + 3

The correct answer is the expression that results from simplifying both terms and combining them. Simplifying the second term of the expression yields

\frac{x^2+x}{x-1}

. Subtracting this from the first term gives

\frac{2x^2+3x-3 - (x^2+x)}{x-1} = \frac{x^2+2x-3}{x-1}

. Factoring the numerator gives

\frac{(x+3)(x-1)}{x-1}

, which simplifies to the expression representing the sum of

x

and

3

since

x-1

cancels out.

Step-by-Step Solution

1

Factor the numerator and denominator of the second term of the expression.

The numerator is

x^3 + 2x^2 + x = x(x^2 + 2x + 1) = x(x+1)^2

. The denominator is

x^2 - 1 = (x-1)(x+1)

.

Factoring allows for the simplification of the rational expression by identifying common factors.

2

Simplify the second term by canceling the common factor

(x+1)

since

x > 1

.

The second term simplifies to

\frac{x(x+1)}{x-1} = \frac{x^2+x}{x-1}

.

Since

x > 1

,

x+1 \neq 0

, so we can cancel it from both numerator and denominator.

3

Subtract the simplified second term from the first term.

\frac{2x^2 + 3x - 3}{x - 1} - \frac{x^2 + x}{x - 1} = \frac{(2x^2 + 3x - 3) - (x^2 + x)}{x - 1} = \frac{x^2 + 2x - 3}{x - 1}

.

With a common denominator of

x-1

, we can combine the numerators directly, being careful to distribute the subtraction sign to both terms in the second numerator.

4

Factor the quadratic expression in the numerator.

x^2 + 2x - 3 = (x+3)(x-1)

.

Factoring the quadratic helps to check if there is a common factor with the denominator.

5

Cancel the common factor

(x-1)

to find the final simplified expression.

\frac{(x+3)(x-1)}{x-1} = x+3

.

Since

x > 1

,

x-1 \neq 0

, allowing us to divide out the common factor.

Key Concept

Simplifying rational expressions and combining algebraic terms over a common denominator

Alternative Method

Instead of simplifying the second term first, a common denominator can be established by multiplying the first term by

\frac{x+1}{x+1}

. This gives:

\frac{(2x^2+3x-3)(x+1) - (x^3+2x^2+x)}{x^2-1} = \frac{2x^3+5x^2-3 - x^3-2x^2-x}{x^2-1} = \frac{x^3+3x^2-x-3}{x^2-1}

. Factoring by grouping the numerator yields

\frac{(x^2-1)(x+3)}{x^2-1} = x+3

.

Estimated Time:2m 0s

Question 2Question →

If the expression $\frac{3x^2 - bx + 10}{x - 2}$ is equivalent to $3x - c - \frac{12}{x - 2}$ for all $x \neq 2$ , where $b$ and $c$ are positive constants, what is the value of $b$ ?

Show answer & explanation

Answer: 17

Answer

17

Multiplying both sides of the equivalence by

x - 2

results in

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

. Expanding the right side gives

3x^2 - (6 + c)x + (2c - 12)

. By equating the coefficients of corresponding terms, we get

2c - 12 = 10

, which solves to

c = 11

. Substituting this into the

x

-coefficient equivalence

-b = -(6 + c)

gives

b = 6 + 11 = 17

.

Step-by-Step Solution

1

Multiply both sides of the expression by the denominator

x - 2

to equate the numerators.

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

Since the rational expressions are equivalent for all

x \neq 2

, their numerators must be equal when written over a common denominator.

2

Expand and simplify the right side of the equation.

3x^2 - bx + 10 = 3x^2 - (6 + c)x + (2c - 12)

Expanding

(3x - c)(x - 2)

yields

3x^2 - 6x - cx + 2c

, and subtracting

12

gives the simplified polynomial expression.

3

Equate the constant terms to solve for

c

.

10 = 2c - 12 \Rightarrow c = 11

For the polynomials to be equivalent, their constant terms must be equal.

4

Equate the coefficients of the

x

terms to solve for

b

.

b = 6 + c \Rightarrow b = 6 + 11 = 17

Equating the coefficients of

x

gives

-b = -(6 + c)

, which simplifies to

b = 6 + c

.

Key Concept

Equivalence of rational and polynomial expressions via coefficient comparison

Alternative Method

Alternatively, you can evaluate the equivalence at a convenient value of

x

. For instance, substituting

x = 0

into the expression gives

-5 = -c + 6

, yielding

c = 11

. Then, evaluating the equation at another convenient value such as

x = 1

allows you to solve for

b

directly using the now-known value of

c

.

Estimated Time:1m 30s

Question 3Question →

If the expression $(x + 4)^2 - (x - 4)^2$ is equivalent to $kx$ for all values of $x$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of the constant

k

is

16

.

To find the value of

k

, the expression

(x + 4)^2 - (x - 4)^2

is simplified by expanding both binomials. The first binomial expands to

x^2 + 8x + 16

, and the second binomial expands to

x^2 - 8x + 16

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x - (-8x) + 16 - 16 = 16x

. Equating

16x

to

kx

yields

k = 16

.

Step-by-Step Solution

1

Expand the first squared term,

(x + 4)^2

x^2 + 8x + 16

Applying the binomial expansion formula

(a + b)^2 = a^2 + 2ab + b^2

where

a = x

and

b = 4

.

2

Expand the second squared term,

(x - 4)^2

x^2 - 8x + 16

Applying the binomial expansion formula

(a - b)^2 = a^2 - 2ab + b^2

where

a = x

and

b = 4

.

3

Subtract the second expanded expression from the first and simplify

16x

Group like terms and distribute the negative sign:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x + 8x + 16 - 16 = 16x

.

4

Equate the simplified expression to

kx

to determine the constant

k

k = 16

Since

16x = kx

for all values of

x

, the coefficients must be equal.

Key Concept

Equivalent Algebraic Expressions

Question 4Question →

For all $x > 0$ , the expression $\frac{\sqrt{x^7}}{\sqrt[5]{x^3}}$ is equivalent to $x^a$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2.9

Answer

2.9

To rewrite the expression, we convert the radicals to rational exponents:

\sqrt{x^7} = x^{7/2}

and

\sqrt[5]{x^3} = x^{3/5}

. Using the quotient property of exponents,

\frac{x^m}{x^n} = x^{m-n}

, we subtract the denominator's exponent from the numerator's exponent:

\frac{7}{2} - \frac{3}{5}

. Finding a common denominator of 10 gives

\frac{35}{10} - \frac{6}{10} = \frac{29}{10}

, which simplifies to

2.9

. Thus, the value of

a

is

2.9

.

Step-by-Step Solution

1

Convert the radical expressions to rational exponents.

Numerator:

\sqrt{x^7} = x^{7/2}

; Denominator:

\sqrt[5]{x^3} = x^{3/5}

.

Radical expressions

\sqrt[n]{x^m}

can be rewritten as rational exponent expressions

x^{m/n}

.

2

Apply the quotient rule of exponents.

x^{7/2} / x^{3/5} = x^{7/2 - 3/5}

.

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Subtract the exponents by finding a common denominator.

\frac{7}{2} - \frac{3}{5} = \frac{35}{10} - \frac{6}{10} = \frac{29}{10} = 2.9

.

Subtracting the fractions requires a common denominator of 10. The result

\frac{29}{10}

is equivalent to the terminating decimal

2.9

.

Key Concept

Equivalent algebraic expressions involving radical and rational exponents

Question 5Question →

For all $x > 0$ , the expression $\frac{6x^2 + 13x - 5}{2x + 5} - \frac{3x^2 - 4x}{x}$ is equivalent to a constant $c$ . What is the value of $c$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

To find the constant value of the expression, both rational expressions are first simplified. The quadratic in the numerator of the first fraction,

6x^2 + 13x - 5

, can be factored as

(3x - 1)(2x + 5)

. Canceling the common factor of

2x + 5

in the numerator and denominator simplifies the first term to

3x - 1

. The second fraction,

\frac{3x^2 - 4x}{x}

, can be simplified by dividing each term in the numerator by

x

, which results in

3x - 4

. Subtracting the second simplified expression from the first gives

(3x - 1) - (3x - 4) = 3x - 1 - 3x + 4 = 3

. Thus, the constant value of the expression is

3

.

Step-by-Step Solution

1

Factor the numerator of the first rational expression.

6x^2 + 13x - 5 = (3x - 1)(2x + 5)

This allows us to simplify the fraction by canceling common factors.

2

Simplify the first fraction by canceling the common term

(2x + 5)

.

\frac{(3x - 1)(2x + 5)}{2x + 5} = 3x - 1

Since

x > 0

,

2x + 5 \neq 0

, so we can cancel the common factor.

3

Simplify the second fraction by factoring out

x

and canceling it.

\frac{3x^2 - 4x}{x} = 3x - 4

Since

x > 0

,

x \neq 0

, so we can divide each term in the numerator by

x

.

4

Subtract the two simplified expressions.

(3x - 1) - (3x - 4) = 3

To find the constant value

c

, we subtract the second simplified expression from the first and combine like terms, paying close attention to distributing the negative sign.

Key Concept

Simplifying rational expressions by factoring and performing polynomial subtraction.

Question 6Question →

For all $x \ge 0$ , the expression $(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)$ is equivalent to $ax - b\sqrt{x} - 10$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

17

Expanding the expression

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

yields

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10

. Combining like terms gives

6x - 11\sqrt{x} - 10

. Comparing this result to the given equivalent form

ax - b\sqrt{x} - 10

shows that

a = 6

and

b = 11

. Therefore, the value of

a + b

is

6 + 11 = 17

.

Step-by-Step Solution

1

Multiply the binomials

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

using FOIL.

(3x^{\frac{1}{2}})(2x^{\frac{1}{2}}) - 5(3x^{\frac{1}{2}}) + 2(2x^{\frac{1}{2}}) - 10

Expanding the product allows us to write it in the standard simplified form.

2

Simplify the products and combine the like terms.

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10 = 6x - 11x^{\frac{1}{2}} - 10

This simplifies the expression so we can compare it directly to the target form.

3

Rewrite

x^{\frac{1}{2}}

as

\sqrt{x}

and compare the expression to

ax - b\sqrt{x} - 10

.

6x - 11\sqrt{x} - 10 = ax - b\sqrt{x} - 10

, which gives

a = 6

and

b = 11

.

Matching corresponding coefficients allows us to determine the values of the constants

a

and

b

.

4

Calculate the sum of

a

and

b

.

a + b = 6 + 11 = 17

The question requires finding the value of

a + b

.

Key Concept

To determine equivalence between algebraic expressions, expand the terms using the distributive property, simplify, and equate the corresponding coefficients of the like terms.

Question 7Question →

Which expression is equivalent to $3(2x - 5) - (x - 4)$ ?

5x - 19

5x - 11

5x - 9

5x - 1

Show answer & explanation

Answer: $5x - 11$

Answer

5x - 11

To find the equivalent expression, apply the distributive property to both parts of the expression:

3(2x - 5)

expands to

6x - 15

, and

-(x - 4)

expands to

-x + 4

. Combining the variable terms gives

6x - x = 5x

. Combining the constant terms gives

-15 + 4 = -11

. Thus, the simplified equivalent expression is

5x - 11

.

Step-by-Step Solution

1

Distribute the number

3

to each term inside the first set of parentheses.

3(2x - 5) = 6x - 15

To eliminate the first set of parentheses by applying the distributive property.

2

Distribute the negative sign (equivalent to multiplying by

-1

) to each term inside the second set of parentheses.

-(x - 4) = -x + 4

To eliminate the second set of parentheses, ensuring the negative sign is applied to both

x

and

-4

.

3

Combine the like terms (the variable terms and the constant terms).

(6x - x) + (-15 + 4) = 5x - 11

To simplify the expression into its final equivalent form.

Key Concept

Equivalent Algebraic Expressions

Question 8Question →

If the expression $(2x + 3)(3x + 1) - 6x^2$ is rewritten in the form $bx + c$ , where $b$ and $c$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: 11

Answer

The value of

b

is

11

.

Expanding the expression

(2x + 3)(3x + 1) - 6x^2

yields

(6x^2 + 2x + 9x + 3) - 6x^2

. Combining the linear terms simplifies the expression to

(6x^2 + 11x + 3) - 6x^2

, which further simplifies to

11x + 3

. Comparing this to the form

bx + c

, we see that the coefficient

b

of the

x

term is

11

.

Step-by-Step Solution

1

Expand the product of the binomials

(2x + 3)(3x + 1)

using the distributive property.

6x^2 + 2x + 9x + 3

To rewrite the factored part of the expression in polynomial form.

2

Combine the linear terms

2x

and

9x

.

6x^2 + 11x + 3

To simplify the expanded polynomial expression.

3

Subtract

6x^2

from the simplified polynomial expression.

11x + 3

To complete the subtraction indicated in the original expression.

4

Compare the resulting expression

11x + 3

to the form

bx + c

to identify the coefficient

b

.

b = 11

To determine the constant coefficient of the

x

term.

Key Concept

Expanding products of binomials and combining like terms to find equivalent expressions.

Question 9Question →

Which of the following expressions is equivalent to $(2x^3)^4$ for all values of $x$ ?

16x^7

8x^{12}

16x^{12}

8x^7

Show answer & explanation

Answer: $16x^{12}$

Answer

16x^{12}

The correct answer is the expression

16x^{12}

. When simplifying

(2x^3)^4

, we apply the exponent

4

to both the coefficient

2

and the term

x^3

, resulting in

2^4(x^3)^4

. Evaluating

2^4

gives

16

. Applying the power of a power rule to

(x^3)^4

requires multiplying the exponents (

3 \times 4

), which gives

x^{12}

. Combining these yields

16x^{12}

.

Step-by-Step Solution

1

Apply the power of a product property to distribute the outer exponent of

4

to both factors inside the parentheses: the coefficient

2

and the variable term

x^3

.

(2x^3)^4 = 2^4 \times (x^3)^4

The power of a product rule states that

(ab)^n = a^n b^n

.

2

Evaluate the constant coefficient

2^4

.

2^4 = 2 \times 2 \times 2 \times 2 = 16

Evaluating the base raised to the exponent.

3

Apply the power of a power property to simplify the variable term

(x^3)^4

.

(x^3)^4 = x^{3 \times 4} = x^{12}

The power of a power rule states that

(x^a)^b = x^{a \times b}

.

4

Combine the simplified coefficient and variable parts to find the final equivalent expression.

16x^{12}

Combining the results of the coefficient simplification and the variable simplification yields the final term.

Key Concept

Exponent rules, specifically the power of a product property

(ab)^n = a^n b^n

and the power of a power property

(x^a)^b = x^{a \times b}

.

Estimated Time:45s

Question 10Question →

If the expression $5(x^2 - 2x) - 3(x^2 - 4x)$ is rewritten in the form $ax^2 + bx$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of

a

is

2

.

Distributing the constants outside the parentheses yields

5x^2 - 10x - 3x^2 + 12x

. Combining the

x^2

terms (

5x^2 - 3x^2 = 2x^2

) and the

x

terms (

-10x + 12x = 2x

) gives the simplified equivalent expression

2x^2 + 2x

. Comparing this directly to the form

ax^2 + bx

, we find that the coefficient

a

is

2

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

5x^2 - 10x - 3x^2 + 12x

To expand the expression so that like terms can be combined.

2

Combine the coefficients of

x^2

and the coefficients of

x

.

2x^2 + 2x

To write the expression in simplified form.

3

Identify the value of

a

by matching the simplified expression with

ax^2 + bx

.

a = 2

The constant

a

represents the coefficient of the

x^2

term.

Key Concept

Simplifying polynomial expressions by distributing constants and combining like terms.

Estimated Time:45s

Question 11Question →

Which of the following expressions is equivalent to $(8x^6)^{\frac{2}{3}} \cdot (4x^{-2})^{-\frac{1}{2}}$ for all positive values of $x$ ?

2x^3

2x^4

2x^5

8x^5

Show answer & explanation

Answer: $2x^5$

Answer

2x^5

To find the equivalent expression, simplify each part of the product. The first part,

(8x^6)^{\frac{2}{3}}

, simplifies to

8^{\frac{2}{3}} \cdot x^{6 \cdot \frac{2}{3}} = 4x^4

. The second part,

(4x^{-2})^{-\frac{1}{2}}

, simplifies to

4^{-\frac{1}{2}} \cdot x^{-2 \cdot -\frac{1}{2}} = \frac{1}{2}x

. Multiplying these two simplified expressions gives

4x^4 \cdot \frac{1}{2}x = 2x^5

. This matches the expression

2x^5

.

Step-by-Step Solution

1

Simplify the first term,

(8x^6)^{\frac{2}{3}}

, by distributing the exponent to both the coefficient and the variable.

(8x^6)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot (x^6)^{\frac{2}{3}} = (\sqrt[3]{8})^2 \cdot x^{6 \cdot \frac{2}{3}} = 2^2 \cdot x^4 = 4x^4

The power of a product rule states that

(ab)^n = a^n b^n

, and the power of a power rule states that

(a^m)^n = a^{mn}

.

2

Simplify the second term,

(4x^{-2})^{-\frac{1}{2}}

, by distributing the exponent to both the coefficient and the variable.

(4x^{-2})^{-\frac{1}{2}} = 4^{-\frac{1}{2}} \cdot (x^{-2})^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} \cdot x^{-2 \cdot -\frac{1}{2}} = \frac{1}{2}x^1 = \frac{1}{2}x

Applying the same power rules, negative exponent properties, and multiplying negative exponents yields a positive exponent:

-2 \cdot -\frac{1}{2} = 1

.

3

Multiply the simplified expressions obtained in Step 1 and Step 2.

4x^4 \cdot \frac{1}{2}x = (4 \cdot \frac{1}{2}) \cdot (x^4 \cdot x^1) = 2x^{4+1} = 2x^5

Multiply the coefficients and add the exponents of the same base variable according to the product rule

a^m \cdot a^n = a^{m+n}

.

Key Concept

Simplifying algebraic expressions containing rational and negative exponents by using the rules of exponents.

Question 12Question →

For $x > 0$ , which of the following is equivalent to the expression $\frac{\sqrt{x^3 \cdot \sqrt{x}}}{\sqrt[4]{x^3}}$ ?

x^4

x\sqrt[4]{x}

x

x^2\sqrt{x}

Show answer & explanation

Answer: $x$

Answer

The expression is equivalent to

x

.

To simplify the expression, convert each radical into an expression with a rational exponent. The inner term of the numerator contains

\sqrt{x} = x^{1/2}

. Multiplying this by

x^3

yields

x^3 \cdot x^{1/2} = x^{3 + 1/2} = x^{7/2}

. Applying the outer square root to this product yields

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

. The denominator is

\sqrt[4]{x^3} = x^{3/4}

. Dividing the simplified numerator by the denominator is performed by subtracting the exponent of the denominator from that of the numerator:

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

. Therefore, the expression simplifies to

x

.

Step-by-Step Solution

1

Convert the inner radical in the numerator into a fractional exponent.

The term

\sqrt{x}

becomes

x^{1/2}

, so the product inside the outer square root becomes

x^3 \cdot x^{1/2}

.

Converting radicals to rational exponents allows the application of standard exponent properties.

2

Combine the terms inside the outer square root using exponent addition rules.

Since

3 + 1/2 = 7/2

, the product is

x^{7/2}

.

When multiplying exponential terms with the same base, their exponents are added.

3

Simplify the entire numerator by applying the outer square root.

The numerator becomes

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

.

Applying a square root to a term is equivalent to raising that term to the power of

1/2

.

4

Convert the denominator into a fractional exponent.

The denominator

\sqrt[4]{x^3}

becomes

x^{3/4}

.

Converting the denominator's radical to a rational exponent allows division with the numerator.

5

Divide the numerator by the denominator using exponent subtraction rules.

The final expression is

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

.

When dividing exponential terms with the same base, the exponent of the denominator is subtracted from the exponent of the numerator.

Key Concept

Simplification of equivalent algebraic expressions involving radical and rational exponents.

Alternative Method

An alternative approach is to substitute a convenient value for

x

that is a perfect power of 4, such as

x = 16

. The numerator becomes

\sqrt{16^3 \cdot \sqrt{16}} = \sqrt{4096 \cdot 4} = \sqrt{16384} = 128

. The denominator becomes

\sqrt[4]{16^3} = \sqrt[4]{4096} = 8

. Evaluating the fraction yields

128 / 8 = 16

. Since substituting

x = 16

yields a value of

16

, the expression must be equivalent to

x

.

Estimated Time:1m 15s

Question 13Question →

Which of the following is equivalent to the expression $\frac{(x^2)^3}{x^{-2}}$ for all positive values of $x$ ?

x^3

x^4

x^8

x^7

Show answer & explanation

Answer: $x^8$

Answer

The expression

x^8

is equivalent to the given expression.

The correct expression is

x^8

. First, simplify the numerator using the power of a power rule:

(x^2)^3 = x^{2 \cdot 3} = x^6

. Next, divide by the denominator using the quotient rule of exponents, which requires subtracting the exponent in the denominator from the exponent in the numerator:

x^6 / x^{-2} = x^{6 - (-2)} = x^{6 + 2} = x^8

.

Step-by-Step Solution

1

Apply the power of a power rule to the numerator:

(x^a)^b = x^{a \cdot b}

.

(x^2)^3 = x^{2 \cdot 3} = x^6

To raise an exponential term to a power, multiply the exponents.

2

Apply the quotient rule for exponents:

\frac{x^a}{x^b} = x^{a - b}

.

\frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8

To divide powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Key Concept

Simplifying algebraic expressions containing negative and positive exponents using the rules of exponents.

Question 14Question →

For all positive values of $x$ and $y$ , the expression $\frac{(3x^2 y^4)^3}{9x^4 y^7}$ can be written in the form $a x^b y^c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Show answer & explanation

Answer: 10

Answer

The value of

a + b + c

is

10

.

To find the value of

a + b + c

, we simplify the given expression using the rules of exponents. First, apply the power of a product rule and power of a power rule to the numerator:

(3x^2 y^4)^3 = 3^3 (x^2)^3 (y^4)^3 = 27x^6y^{12}

. Next, divide this by the denominator:

\frac{27x^6y^{12}}{9x^4y^7}

. Divide the coefficients to get

\frac{27}{9} = 3

, and apply the quotient rule for exponents to the variable terms:

x^{6-4} = x^2

and

y^{12-7} = y^5

. The fully simplified expression is

3x^2y^5

, which matches the form

a x^b y^c

. Comparing coefficients and exponents, we find

a = 3

,

b = 2

, and

c = 5

. Summing these values gives

3 + 2 + 5 = 10

.

Step-by-Step Solution

1

Simplify the numerator of the expression using exponent rules.

27x^6y^{12}

Applying the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{mn}

to

(3x^2 y^4)^3

gives

3^3 \cdot x^{2 \times 3} \cdot y^{4 \times 3} = 27x^6y^{12}

.

2

Divide the simplified numerator by the denominator.

3x^2y^5

Divide the coefficients (

27 \div 9 = 3

) and subtract the exponents of the corresponding variables using the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

(

x^{6-4} = x^2

and

y^{12-7} = y^5

).

3

Identify the values of the constants

a

,

b

, and

c

from the simplified expression

3x^2y^5

.

a = 3

,

b = 2

, and

c = 5

Comparing the simplified expression

3x^2y^5

with the template

a x^b y^c

yields

a = 3

,

b = 2

, and

c = 5

.

4

Calculate the sum of

a

,

b

, and

c

.

10

Adding the identified constant values together:

3 + 2 + 5 = 10

.

Key Concept

Simplifying exponential expressions using exponent rules

Question 15Question →

Which of the following expressions is equivalent to $\frac{2x^2 - 3x - 9}{x^2 - 9} - \frac{x}{x+3}$ for all $x > 3$ ?

1

\frac{x^2 + 6x - 9}{x^2 - 9}

\frac{x^2 - 6x - 9}{x^2 - 9}

\frac{3x+3}{x+3}

Show answer & explanation

Answer: $1$

Answer

1

The correct answer is

1

. Factoring the first term gives

\frac{(2x+3)(x-3)}{(x+3)(x-3)}

. Since

x > 3

, we cancel the common non-zero factor of

x-3

to obtain

\frac{2x+3}{x+3}

. Subtracting

\frac{x}{x+3}

from this result yields

\frac{2x+3-x}{x+3} = 1

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first rational expression.

The numerator

2x^2 - 3x - 9

factors as

(2x+3)(x-3)

and the denominator

x^2 - 9

factors as

(x+3)(x-3)

.

Factoring allows us to identify and divide out common factors to simplify the expression.

2

Simplify the first rational expression by cancelling the common factor.

For all

x > 3

, the factor

x-3

is non-zero, so the expression

\frac{(2x+3)(x-3)}{(x+3)(x-3)}

simplifies to

\frac{2x+3}{x+3}

.

This simplifies the subtraction by reducing the first fraction.

3

Subtract the second expression from the simplified first expression.

\frac{2x+3}{x+3} - \frac{x}{x+3} = \frac{(2x+3) - x}{x+3} = \frac{x+3}{x+3} = 1

.

Since the denominators are identical, the numerators can be combined directly.

Key Concept

Simplifying rational expressions by factoring and performing algebraic operations.

Alternative Method

Instead of simplifying the first fraction first, find a common denominator immediately by multiplying the numerator and denominator of the second fraction by

x-3

. This yields:

\frac{2x^2 - 3x - 9}{x^2 - 9} - \frac{x(x-3)}{x^2 - 9} = \frac{2x^2 - 3x - 9 - x^2 + 3x}{x^2 - 9} = \frac{x^2 - 9}{x^2 - 9} = 1

.

Estimated Time:2m 0s

Question 16Question →

For all $x > 2$ , which of the following is equivalent to the expression $\frac{6x^3 - 7x^2 - 16x + 12}{2x^2 - 3x - 2}$ ?

3x + 1 - \frac{7}{2x + 1}

3x + 1 - \frac{5}{2x + 1}

3x - 1 - \frac{5}{2x + 1}

3x - 8 + \frac{14}{2x - 1}

Show answer & explanation

Answer: 3x + 1 - \frac{7}{2x + 1}

Answer

The correct answer is the expression that simplifies to

3x + 1 - \frac{7}{2x + 1}

.

The correct answer is found by factoring the quadratic denominator into

(2x + 1)(x - 2)

. Testing the root

x = 2

in the numerator reveals that

(x - 2)

is also a factor of the numerator, allowing the expression to be rewritten as

(x - 2)(6x^2 + 5x - 6)

. Canceling the common factor

(x - 2)

leaves

\frac{6x^2 + 5x - 6}{2x + 1}

. Performing polynomial long division on this remaining term yields a quotient of

3x + 1

and a remainder of

-7

, which is expressed as the correct answer.

Step-by-Step Solution

1

Factor the denominator of the rational expression.

2x^2 - 3x - 2 = (2x + 1)(x - 2)

Identifying the factors of the denominator helps reveal potential common factors in the numerator.

2

Factor the numerator

6x^3 - 7x^2 - 16x + 12

by testing

x = 2

as a possible root.

Since

6(2)^3 - 7(2)^2 - 16(2) + 12 = 48 - 28 - 32 + 12 = 0

,

(x - 2)

is a factor. Dividing the cubic polynomial by

(x - 2)

yields

6x^3 - 7x^2 - 16x + 12 = (x - 2)(6x^2 + 5x - 6)

.

Finding the common linear factor allows us to simplify the rational expression.

3

Simplify the rational expression by canceling the common factor

(x - 2)

.

For

x > 2

,

\frac{(x - 2)(6x^2 + 5x - 6)}{(x - 2)(2x + 1)} = \frac{6x^2 + 5x - 6}{2x + 1}

.

Since

x > 2

,

x - 2 \neq 0

, which mathematically permits dividing out the common factor.

4

Perform polynomial long division on

\frac{6x^2 + 5x - 6}{2x + 1}

.

6x^2 + 5x - 6 = (3x + 1)(2x + 1) - 7

.

Dividing the quadratic expression by the linear expression separates the polynomial into a quotient and a rational remainder.

5

Rewrite the final expression with the quotient and remainder.

3x + 1 - \frac{7}{2x + 1}

Expressing the result in the standard quotient-remainder form matches the target expression.

Key Concept

Simplification and division of rational algebraic expressions

Question 17Question →

If the expression $\frac{3x^3 - 8x^2 + kx - 6}{3x - 2}$ is equivalent to $x^2 - 2x + 3$ for all $x \neq \frac{2}{3}$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 13

Answer

13

By multiplying both sides of the equation by

3x - 2

, the rational expression simplifies to a polynomial identity:

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

. Expanding the right side yields

3x^3 - 8x^2 + 13x - 6

. Since the two polynomials are equivalent, their corresponding coefficients must be equal, meaning the coefficient of the linear term,

k

, must be equal to

13

.

Step-by-Step Solution

1

Multiply both sides of the equivalence by the denominator

(3x - 2)

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

To clear the fraction and align the polynomial expressions for coefficient comparison.

2

Expand the right side of the equation using the distributive property

3x^3 - 8x^2 + 13x - 6

To obtain the expanded form of the polynomial so that we can identify the coefficients of each term.

3

Equate the corresponding coefficients of the linear

x

terms on both sides of the equation

k = 13

Since the two expressions are equivalent for all values of

x

, their coefficients for each corresponding power of

x

must be equal.

Key Concept

Equivalent Algebraic Expressions

Question 18Question →

Which of the following is equivalent to the expression $\frac{2x^2 - 7x - 4}{x^2 - 16} \cdot \frac{x^2 + 4x}{2x + 1}$ for all values of $x$ where the expression is defined?

\frac{x(2x-1)}{2x+1}

x

x^2

\frac{x(x+4)}{x-4}

Show answer & explanation

Answer: $x$

Answer

x

The correct answer is

x

because factoring all numerators and denominators of the given product results in

\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1}

. After cancelling the common factors of

(2x + 1)

,

(x - 4)

, and

(x + 4)

, only

x

remains.

Step-by-Step Solution

1

Factor the quadratic expression in the numerator of the first fraction.

2x^2 - 7x - 4 = (2x + 1)(x - 4)

Finding the factors of the quadratic trinomial allows us to look for common terms to cancel later.

2

Factor the denominator of the first fraction using the difference of squares identity.

x^2 - 16 = (x - 4)(x + 4)

Simplifying the quadratic binomial in the denominator exposes common factors.

3

Factor the numerator of the second fraction by extracting the greatest common factor.

x^2 + 4x = x(x + 4)

Exhibiting the shared variable

x

helps identify terms that will cancel with the denominator.

4

Substitute the factored forms back into the original expression and cancel the common factors.

\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1} = x

The terms

(2x + 1)

,

(x - 4)

, and

(x + 4)

appear in both the numerator and the denominator, leaving only the factor

x

.

Key Concept

Simplifying rational expressions by factoring and cancelling common factors

Estimated Time:1m 30s

Question 19Question →

Which of the following expressions is equivalent to $3(x^2 - 2x) - 2(x^2 - 5x)$ for all values of $x$ ?

x^2 - 16x

x^2 + 4x

x^2 - 4x

5x^2 - 16x

Show answer & explanation

Answer: $x^2 + 4x$

Answer

x^2 + 4x

To simplify the given expression, distribute the coefficients outside the parentheses to each term inside. Distributing

3

to

(x^2 - 2x)

yields

3x^2 - 6x

. Distributing

-2

to

(x^2 - 5x)

yields

-2x^2 + 10x

. Grouping and combining the like terms gives

(3x^2 - 2x^2) + (-6x + 10x)

, which simplifies to

x^2 + 4x

.

Step-by-Step Solution

1

Distribute the number

3

to the terms in the first set of parentheses.

3(x^2 - 2x) = 3x^2 - 6x

To eliminate the parentheses by multiplying each term inside by the outer coefficient.

2

Distribute the number

-2

to the terms in the second set of parentheses.

-2(x^2 - 5x) = -2x^2 + 10x

To eliminate the second set of parentheses, remembering that multiplying two negative numbers yields a positive product:

-2 \times -5x = 10x

.

3

Combine like terms by grouping the quadratic terms and the linear terms together.

(3x^2 - 2x^2) + (-6x + 10x) = x^2 + 4x

To simplify the expression by adding or subtracting the coefficients of terms with the same variable power.

Key Concept

Equivalent Algebraic Expressions

Estimated Time:45s

Question 20Question →

If the expression $\frac{6x^3 + 7x^2 - 14x + 19}{2x + 5}$ is equivalent to $ax^2 + bx + c + \frac{k}{2x + 5}$ for all $x \neq -2.5$ , where $a$ , $b$ , $c$ , and $k$ are constants, what is the value of $a - b + c + k$ ?

Show answer & explanation

Answer: 14

Answer

14

Dividing the numerator

6x^3 + 7x^2 - 14x + 19

by the denominator

2x + 5

using polynomial division yields a quotient of

3x^2 - 4x + 3

and a remainder of

4

. The equivalent expression is

3x^2 - 4x + 3 + \frac{4}{2x + 5}

. Comparing this with the form

ax^2 + bx + c + \frac{k}{2x + 5}

gives the values

a = 3

,

b = -4

,

c = 3

, and

k = 4

. Substituting these values into

a - b + c + k

gives

3 - (-4) + 3 + 4 = 14

.

Step-by-Step Solution

1

Perform the first step of polynomial long division by dividing

6x^3

by

2x

.

Quotient term:

3x^2

; Remainder:

-8x^2 - 14x + 19

To eliminate the highest-degree term of the numerator.

2

Perform the second step of division by dividing

-8x^2

by

2x

.

Quotient term:

-4x

; Remainder:

6x + 19

To find the next term of the quotient.

3

Perform the third step of division by dividing

6x

by

2x

.

Quotient term:

3

; Remainder:

4

To find the constant term of the quotient and the final remainder.

4

Compare the resulting expression

3x^2 - 4x + 3 + \frac{4}{2x + 5}

with the given form to identify the constants

a

,

b

,

c

, and

k

.

a = 3

,

b = -4

,

c = 3

,

k = 4

To map the coefficients of equivalent algebraic expressions.

5

Evaluate the expression

a - b + c + k

using the identified values.

3 - (-4) + 3 + 4 = 14

To obtain the final numeric value requested.

Key Concept

Equivalent Algebraic Expressions via Polynomial Long Division

Equivalent Algebraic Expressions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Answer

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Common Mistakes

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