Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 9

Question 161Question →

Consider the exponential expression $(y^7)^3$ . If this expression is rewritten in the equivalent form $y^n$ , what is the numerical value of $n$ ?

Show answer & explanation

Answer: 21; twenty-one; n=21; n = 21

Answer

The numerical value of the exponent is 21.

When raising a power to another power, the laws of exponents dictate that the two exponents must be multiplied. Multiplying the inside exponent 7 by the outside exponent 3 yields 21.

Step-by-Step Solution

1

Identify the appropriate exponent rule to simplify the expression

(y^7)^3

.

The expression requires the Power of a Power property.

An exponential term with a base and an exponent is being raised entirely to another power.

2

Apply the Power of a Power rule, which states that

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

.

Multiply the inner exponent by the outer exponent:

7 \times 3

.

To raise a power to another power, you must multiply the exponents together.

3

Calculate the final numerical value.

7 \times 3 = 21

, meaning the expression simplifies to

y^{21}

.

By matching

y^{21}

to the given form

y^n

, we determine that

n = 21

.

Key Concept

Power of a Power Rule for Exponents

Estimated Time:30s

Question 162Question →

Evaluate the product of the monomial expressions $4z^5$ and $2z^3$ . Which of the following represents the simplified result?

8z^8

8z^{15}

6z^8

6z^{15}

Show answer & explanation

Answer: $8z^8$

Answer

The expression simplifies to

8z^8

.

When multiplying two monomial expressions, the numerical coefficients are multiplied together (

4 \times 2 = 8

), and the exponents of the common variable base are added together according to the product rule for exponents (

5 + 3 = 8

). This results in

8z^8

.

Step-by-Step Solution

1

Identify the operation to be performed between the two terms.

The terms

4z^5

and

2z^3

are being multiplied.

Understanding the operation is required to apply the correct mathematical rules for coefficients and exponents.

2

Multiply the numerical coefficients.

4 \times 2 = 8

Coefficients in a product of monomials are multiplied together normally.

3

Multiply the variable parts by applying the product rule for exponents.

z^5 \times z^3 = z^{5+3} = z^8

When multiplying exponential expressions with the same base, their exponents must be added.

4

Combine the calculated coefficient and the variable part.

8z^8

This brings the simplified pieces together into the final term.

Key Concept

Operations with Variable Exponents

Question 163Question →

Consider the polynomial expression provided below:

(2x - 3)(x + 4) - 2(x^2 - 5x + 1)

Which of the following represents its fully simplified equivalent?

10x - 14

-5x - 14

15x - 14

x

Show answer & explanation

Answer: $15x - 14$

Answer

The completely simplified equivalent expression is

15x - 14

.

The expression correctly expands to

2x^2 + 5x - 12

on the left and

-2x^2 + 10x - 2

on the right. When adding these together, the

x^2

terms cancel out, the

x

terms sum to

15x

, and the constants sum to

-14

, leaving

15x - 14

.

Step-by-Step Solution

1

Expand

(2x - 3)(x + 4)

using the distributive property.

2x^2 + 8x - 3x - 12

, which simplifies to

2x^2 + 5x - 12

.

To eliminate the first set of parentheses by multiplying each term in the first binomial by each term in the second binomial.

2

Distribute

-2

to each term inside the trinomial

(x^2 - 5x + 1)

.

-2x^2 + 10x - 2

.

To eliminate the second set of parentheses, carefully applying the negative sign to all inner terms.

3

Combine the expanded expressions.

(2x^2 + 5x - 12) + (-2x^2 + 10x - 2)

.

To prepare the entire expression for the collection of like terms.

4

Group and combine like terms (

x^2

terms,

x

terms, and constants).

(2x^2 - 2x^2) + (5x + 10x) + (-12 - 2) = 0x^2 + 15x - 14 = 15x - 14

.

To achieve the final simplified standard form.

Key Concept

Expanding polynomials, distributing negative coefficients, and combining like terms.

Question 164Question →

Complete the algebraic equation that accurately translates the given statement.

Fill in the blanks below

Statement: 'Five subtracted from four times a number

k

equals eleven.'

Equation: - = 11

Show answer & explanation

Answer

The first blank is '4k' and the second blank is '5', forming the correct equation 4k - 5 = 11.

The phrase 'four times a number

k

' becomes

4k

. The phrase 'Five subtracted from' means 5 is taken away from

4k

, so the 5 must be placed after the minus sign, forming

4k - 5

. Finally, 'equals eleven' sets the expression equal to 11. Therefore, to match the provided equation structure of '___ - ___ = 11', the first blank must be filled with '4k' and the second blank must be filled with '5'.

Step-by-Step Solution

1

Translate the multiplication part of the statement.

'four times a number

k

' translates to the algebraic expression

4k

.

The word 'times' indicates multiplication between the coefficient 4 and the variable

k

.

2

Apply the subtraction part of the statement.

'Five subtracted from' means 5 is taken away from the previous term, resulting in

4k - 5

.

The phrase 'subtracted from' is a turnaround phrase, which means the 5 must be placed after the minus sign, reversing the spoken order.

3

Complete the equation.

Setting the expression equal to 11 gives the full equation:

4k - 5 = 11

.

The word 'equals' indicates where the equal sign (=) should be placed in relation to the constant 11.

Key Concept

Translating turnaround words (like 'subtracted from' or 'less than') and basic multiplication phrases into an algebraic equation.

Estimated Time:30s

Question 165Question →

Which of the following describes all values of $x$ that satisfy the inequality below?

2(x - 4) - 5x < x + 4

x > -3

x < -3

x > 3

x > 1

Show answer & explanation

Answer: $x > -3$

Answer

The correct solution is

x > -3

.

The correct answer accurately simplifies the equation down to

-4x < 12

. Because isolating the variable in the final step requires dividing by

-4

, the less than symbol (

<

) must be reversed to a greater than symbol (

>

), resulting in

x > -3

.

Step-by-Step Solution

1

Distribute the 2 into the parentheses on the left side.

2x - 8 - 5x < x + 4

To clear the parentheses and access the terms inside.

2

Combine the like terms (

2x

and

-5x

) on the left side.

-3x - 8 < x + 4

To simplify the left side of the inequality.

3

Subtract

x

from both sides to group the variable terms.

-4x - 8 < 4

To collect all terms containing

x

on one side.

4

Add 8 to both sides of the inequality.

-4x < 12

To isolate the term with the variable.

5

Divide both sides by -4 and reverse the inequality sign.

x > -3

When multiplying or dividing an inequality by a negative number, the inequality symbol must be flipped to maintain a mathematically true statement.

Key Concept

Solving linear inequalities with variables on both sides requires combining like terms and reversing the inequality symbol when dividing by a negative coefficient.

Question 166Question →

Match each unsimplified algebraic expression on the left with its completely simplified equivalent on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correctly simplified expressions are:

5x - 3(2x - 4)

matches with

-x + 12

;

3(2m - 5n) - 2(m - 3n)

matches with

4m - 9n

;

4x(x - 2) - x^2 + 5x

matches with

3x^2 - 3x

; and

7y - 3[y - 2(y + 1)]

matches with

10y + 6

.

Each unsimplified expression must be correctly expanded using the distributive property, ensuring that negative multipliers are applied to every term inside the parentheses. After expanding, grouping and adding the coefficients of like terms produces the single correct simplified equivalent provided in the match list.

Step-by-Step Solution

1

Distribute any coefficients or negative signs immediately outside of parentheses or brackets.

Terms are expanded (e.g.,

-3(2x - 4)

becomes

-6x + 12

).

Distribution removes grouping symbols, which is required before combining terms.

2

For expressions with nested grouping symbols, simplify from the innermost parentheses outward.

Complex expressions like

7y - 3[y - 2(y + 1)]

simplify step-by-step to

7y + 3y + 6

.

Following the order of operations ensures accurate distribution of multipliers.

3

Group and combine like terms (terms with the exact same variable and exponent).

The final canonical form is reached for each expression.

Combining like terms fully simplifies the polynomial into standard form.

Key Concept

Simplifying Algebraic Expressions

Estimated Time:1m 30s

Question 167Question →

A linear equation is structured as follows:

\frac{1}{2}(4y - 6) - \frac{2}{3}(9 - 6y) = 2y + 15

Based on this relationship, what is the correct value of $y$ ?

-6

-3

3

6

Show answer & explanation

Answer: 6

Answer

6

The correct value is found by properly distributing the fractional coefficients, carefully tracking negative signs, and preserving equality to isolate the variable. The operations simplify the initial equation to 6y - 9 = 2y + 15, which reduces to 4y = 24, resulting in 6.

Step-by-Step Solution

1

Distribute

\frac{1}{2}

into the first set of parentheses.

2y - 3

Applying the distributive property eliminates the first group.

2

Distribute

-\frac{2}{3}

into the second set of parentheses.

-6 + 4y

Multiplying the negative fraction across the terms. Notice that multiplying two negative values (

-2/3

and

-6y

) yields a positive result (

+4y

).

3

Combine the distributed terms on the left side of the equation.

(2y + 4y) + (-3 - 6) = 6y - 9

Combining like terms simplifies the left side of the equation before attempting to isolate the variable.

4

Set the simplified left side equal to the right side and solve for y.

6y - 9 = 2y + 15 \Rightarrow 4y - 9 = 15 \Rightarrow 4y = 24 \Rightarrow y = 6

Using inverse operations groups the variable terms on one side and constant terms on the other to reveal the final value.

Key Concept

Solving complex linear equations involving fractional distribution and sign management.

Question 168Question →

What is the completely simplified form of the algebraic expression $7x - 4(2x - 3y) + 5y$ ?

-x - 7y

-x + 2y

-x + 17y

7x + 4xy + 5y

Show answer & explanation

Answer: $-x + 17y$

Answer

-x + 17y

To completely simplify the expression, you must first distribute the

-4

across the terms in the parentheses. Multiplying

-4

by

2x

gives

-8x

, and multiplying

-4

by

-3y

gives

+12y

(since a negative times a negative is a positive). Substituting this back gives

7x - 8x + 12y + 5y

. Finally, group the like variables:

7x - 8x = -x

, and

12y + 5y = 17y

. Combining these gives

-x + 17y

.

Step-by-Step Solution

1

Distribute the

-4

to both terms inside the parentheses

(2x - 3y)

.

-8x + 12y

The distributive property must be applied first to eliminate the parentheses, paying careful attention to the multiplication of two negative numbers (

-4

and

-3y

).

2

Rewrite the full expression incorporating the expanded terms.

7x - 8x + 12y + 5y

This sets up the expression so that all individual terms are clearly visible for grouping.

3

Combine the like terms for both

x

and

y

.

-x + 17y

Adding the coefficients of identical variables (

7x - 8x = -x

and

12y + 5y = 17y

) provides the expression in its final standard form.

Key Concept

Simplifying Algebraic Expressions via Distributive Property and Like Terms

Question 169Question →

Pair each basic algebraic expression with its correctly simplified result by applying the fundamental laws of exponents.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

m^6 \cdot m^2

with

m^8

;

\frac{n^8}{n^5}

with

n^3

;

(p^4)^3

with

p^{12}

; and

q^{-5}

with

\frac{1}{q^5}

.

Each expression can be simplified by directly applying one specific fundamental property of exponents.

m^6 \cdot m^2

uses the product rule.

\frac{n^8}{n^5}

uses the quotient rule.

(p^4)^3

uses the power rule.

q^{-5}

uses the negative exponent definition.

Step-by-Step Solution

1

Simplify the product expression

m^6 \cdot m^2

.

m^8

When bases are identical and multiplied, their exponents are added together.

2

Simplify the quotient expression

\frac{n^8}{n^5}

.

n^3

When dividing identical bases, the exponent of the denominator is subtracted from the exponent of the numerator.

3

Simplify the nested power expression

(p^4)^3

.

p^{12}

When raising an exponent to another power, the two exponents are multiplied.

4

Rewrite the negative exponent expression

q^{-5}

.

\frac{1}{q^5}

A negative exponent indicates taking the reciprocal of the base raised to the positive exponent.

Key Concept

Applying fundamental exponent rules (product, quotient, power, and negative exponents) to simplify single-variable expressions.

Question 170Question →

Which value of $m$ is the solution to the equation below?

6m - 12 = 2m + 16

7

1

-2

-7

Show answer & explanation

Answer: $7$

Answer

The correct solution is

7

.

By subtracting

2m

from both sides and then adding 12 to both sides, the equation simplifies to

4m = 28

. Dividing by 4 gives the correct result of

7

.

Step-by-Step Solution

1

Subtract

2m

from both sides of the equation to group the variable terms on the left side.

4m - 12 = 16

This keeps all instances of the unknown variable

m

on one side.

2

Add 12 to both sides of the equation to isolate the variable term.

4m = 28

This cancels out the constant on the left side, leaving only the term with

m

.

3

Divide both sides by 4 to solve for

m

.

m = 7

This removes the coefficient, revealing the final value of the variable.

Key Concept

Solving One-Variable Linear Equations

Estimated Time:45s

Question 171Question →

Calculate the value of the unknown variable.

Fill in the blanks below

Calculate the value of

k

that makes the following equation true:

\frac{3}{4}(8k - 12) - \frac{2}{5}(15 - 5k) = 3(k + 2) + 4

k =

Show answer & explanation

Answer

5

The correct value is 5. By distributing the fractions on the left, the expression becomes

6k - 9 - 6 + 2k

. Combining like terms simplifies this to

8k - 15

. On the right side, distributing the 3 results in

3k + 6 + 4

, which simplifies to

3k + 10

. Equating both simplified sides gives

8k - 15 = 3k + 10

. Subtracting

3k

from both sides leaves

5k - 15 = 10

, and adding

15

to both sides yields

5k = 25

. Dividing by 5 results in

k = 5

.

Step-by-Step Solution

1

Distribute the fractional terms on the left side of the equation.

6k - 9 - 6 + 2k

Multiplying

\frac{3}{4}

by

(8k - 12)

yields

6k - 9

. Multiplying

-\frac{2}{5}

by

(15 - 5k)

yields

-6 + 2k

. Paying close attention to the negative sign during distribution is critical.

2

Distribute the terms on the right side of the equation.

3k + 6 + 4

Multiplying

3

by

(k + 2)

expands the expression, setting up the right side for simplification.

3

Combine like terms on both sides of the equal sign.

8k - 15 = 3k + 10

On the left side, combine the variable terms (

6k + 2k = 8k

) and constant terms (

-9 - 6 = -15

). On the right side, combine the constant terms (

6 + 4 = 10

).

4

Isolate the variable

k

to find its value.

5k = 25

, which simplifies to

k = 5

.

Subtract

3k

from both sides to group the variables on the left (

5k - 15 = 10

). Then, add

15

to both sides to group the constants on the right (

5k = 25

). Dividing both sides by

5

yields the final solution.

Key Concept

Solving One-Variable Linear Equations

Question 172Question →

Determine the numerical value of the variable that satisfies the given linear equation.

Fill in the blanks below

For the equation

7k - 5 = 3k + 19

, the value of

k

is .

Show answer & explanation

Answer

The correct value for k is 6.

To find the value of

k

, we must isolate the variable on one side of the equation. By subtracting

3k

from both sides, we get

4k - 5 = 19

. Next, adding

5

to both sides gives

4k = 24

. Finally, dividing both sides by

4

results in

k = 6

.

Step-by-Step Solution

1

Subtract

3k

from both sides of the equation.

4k - 5 = 19

This groups all variable terms on the left side of the equation.

2

Add

5

to both sides of the equation.

4k = 24

This isolates the variable term by moving the constant to the right side.

3

Divide both sides by

4

.

k = 6

This cancels the coefficient on the variable, yielding the final solution.

Key Concept

Solving linear equations with variables on both sides by using inverse operations to isolate the unknown.

Question 173Question →

Which of the following is equivalent to the algebraic expression below after it has been fully simplified?

4k(k - 3m) - (2k^2 - 5km + m^2)

2k^2 - 7km - m^2

2k^2 - 17km + m^2

6k^2 - 7km - m^2

2k^2 - 8km^2

Show answer & explanation

Answer: $2k^2 - 7km - m^2$

Answer

The fully simplified expression is

2k^2 - 7km - m^2

.

The correct expression is derived by first multiplying

4k

by both terms in

(k - 3m)

to get

4k^2 - 12km

. Next, distributing the negative sign through the trinomial changes the signs to yield

-2k^2 + 5km - m^2

. Finally, grouping and combining like terms yields

4k^2 - 2k^2 = 2k^2

and

-12km + 5km = -7km

. The

-m^2

term has no like terms to combine with, leaving it unchanged. The final simplified form is

2k^2 - 7km - m^2

.

Step-by-Step Solution

1

Distribute

4k

into the binomial

(k - 3m)

.

4k^2 - 12km

Expanding the first part of the expression removes the first set of parentheses.

2

Distribute the negative sign across the trinomial

(2k^2 - 5km + m^2)

.

-2k^2 + 5km - m^2

Subtracting a polynomial requires reversing the sign of every term inside the parentheses.

3

Group and combine all like terms.

(4k^2 - 2k^2) + (-12km + 5km) - m^2 = 2k^2 - 7km - m^2

Expressions are fully simplified when all terms with identical variable combinations are merged.

Key Concept

Distributing variables and negative signs followed by combining like terms.

Question 174Question →

If the equation $y^8 \cdot y^k = y^{15}$ is true for all non-zero values of $y$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7; seven

Answer

7

The correct value of

k

is

7

because applying the product rule of exponents gives

y^8 \cdot y^7 = y^{8+7} = y^{15}

.

Step-by-Step Solution

1

Apply the product rule for exponents to the left side of the equation.

y^8 \cdot y^k = y^{8+k}

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

2

Set the exponent of the simplified left side equal to the exponent of the right side.

8 + k = 15

Since the bases are equal and non-zero, their exponents must be equal.

3

Solve the linear equation for

k

.

k = 7

Subtracting 8 from both sides of the equation isolates the variable

k

.

Key Concept

Product Rule for Exponents

Question 175Question →

Simplify the following algebraic expression completely:

(3a^5)^2

What is the simplified form of this expression?

9a^{10}

9a^7

6a^{10}

3a^{10}

Show answer & explanation

Answer: $9a^{10}$

Answer

The simplified expression is

9a^{10}

.

The correct result is found by distributing the exponent of 2 to both factors inside the parentheses. The coefficient 3 is squared to become 9, and the exponent 5 is multiplied by 2 to become 10, resulting in

9a^{10}

.

Step-by-Step Solution

1

Apply the power of a product rule by distributing the outer exponent to both the numerical coefficient and the variable inside the parentheses.

3^2 \cdot (a^5)^2

When a product is raised to a power, each independent factor inside the parentheses must be raised to that power.

2

Simplify the numerical coefficient by calculating its square.

9 \cdot (a^5)^2

The exponent 2 indicates multiplying the base by itself:

3 \times 3 = 9

.

3

Apply the power of a power rule to the variable by multiplying the inner exponent by the outer exponent.

9a^{10}

According to exponent laws,

(x^m)^n = x^{m \cdot n}

, so the new exponent is

5 \times 2 = 10

.

Key Concept

Power of a Product and Power of a Power Rules

Question 176Question →

An inventory manager calculates the total number of pallets, $h$ , that can be stored in a warehouse section using the following linear equation:

3h + 6 = 45

What is the value of $h$ ?

13

5

17

-13

Show answer & explanation

Answer: 13

Answer

The correct value for the variable is 13.

The correct value is found by performing the inverse operations in the correct order. First, subtract 6 from both sides to get 3h = 39. Then, divide both sides by 3 to find that h = 13.

Step-by-Step Solution

1

Isolate the term with the variable by subtracting 6 from both sides of the equation.

3h = 45 - 6

3h = 39

This removes the constant term from the left side, keeping the equation balanced.

2

Solve for

h

by dividing both sides of the equation by 3.

h = \frac{39}{3}

h = 13

Dividing by the coefficient of the variable isolates

h

completely.

Key Concept

Solving basic two-step linear equations by using inverse operations to isolate the variable.

Question 177Question →

Solve the following equation for $p$ :

\frac{2}{3}(6p - 15) + 3p = -3(p - 8) + 2p - 10

What is the final value of $p$ ?

4

2

3

-3

Show answer & explanation

Answer: 3

Answer

The correct value for the variable is 3.

The correct answer is 3. Distributing the terms on both sides yields

4p - 10 + 3p = -3p + 24 + 2p - 10

. Combining like terms results in

7p - 10 = -p + 14

. Adding

p

to both sides and adding 10 to both sides gives

8p = 24

. Dividing by 8 isolates the variable, resulting in

p = 3

.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses on both sides of the equation.

4p - 10 + 3p = -3p + 24 + 2p - 10

To simplify the equation, all terms inside parentheses must be multiplied by their outer coefficients. Note that

\frac{2}{3} \times 6p = 4p

and

\frac{2}{3} \times -15 = -10

.

2

Combine like terms on each side of the equation independently.

7p - 10 = -p + 14

Combining variable terms and constant terms on the same side simplifies the equation structure before isolating the variable.

3

Move all variable terms to one side of the equation.

8p - 10 = 14

Adding

p

to both sides cancels the

-p

on the right side, grouping all unknowns on the left.

4

Isolate the variable by moving constants to the other side and dividing.

8p = 24 \Rightarrow p = 3

Adding 10 to both sides moves the constant, and dividing both sides by the coefficient 8 solves for

p

.

Key Concept

Solving complex one-variable linear equations requiring distribution and combining like terms.

Question 178Question →

Consider the following algebraic equation, where $x \neq 0$ :

\frac{(3x^4)^2}{9x^k} = x^5

What is the value of the exponent $k$ ?

Show answer & explanation

Answer: 3; k=3; k = 3; three

Answer

3

First, the numerator

(3x^4)^2

simplifies to

9x^8

. Substituting this back yields

\frac{9x^8}{9x^k} = x^5

. The coefficients cancel out, and applying the quotient rule to the variables gives

x^{8-k} = x^5

. By equating the exponents, we get

8 - k = 5

, which means

k = 3

.

Step-by-Step Solution

1

Apply the power of a product rule to the numerator.

(3x^4)^2 = 3^2 \cdot (x^4)^2 = 9x^8

When raising a product to a power, you must raise every factor inside the parentheses to that power.

2

Substitute the simplified numerator back into the equation and simplify the numerical coefficients.

\frac{9x^8}{9x^k} = \frac{x^8}{x^k}

The coefficients

\frac{9}{9}

cancel out to 1, making it easier to isolate the variable base.

3

Apply the quotient rule for exponents on the left side of the equation.

\frac{x^8}{x^k} = x^{8-k}

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

4

Set the resulting exponent equal to the exponent on the right side of the original equation and solve for

k

.

8 - k = 5 \Rightarrow k = 3

Since the bases on both sides of the equation are the same (

x

), their exponents must be equal.

Key Concept

Operations with Variable Exponents

Question 179Question →

If $x = -2$ and $y = -3$ , what is the value of the expression $2x^2 - 5y$ ?

-7

7

23

31

Show answer & explanation

Answer: 23

Answer

23

The expression is evaluated by first substituting the values and applying the exponent to the base, giving

(-2)^2 = 4

. Then, perform the multiplication to get

2(4) = 8

and

5(-3) = -15

. Finally, subtracting negative 15 from 8 is the same as adding 15, resulting in

8 + 15 = 23

.

Step-by-Step Solution

1

Substitute the given values for

x

and

y

into the expression.

2(-2)^2 - 5(-3)

To evaluate the expression, each variable must be replaced with its corresponding numerical value.

2

Evaluate the exponent.

2(4) - 5(-3)

According to the order of operations (PEMDAS), exponents must be resolved before multiplication. The square of any non-zero real number is positive, so

(-2)^2 = 4

.

3

Perform the multiplication operations.

8 - (-15)

Multiply

2

by

4

to get

8

, and

5

by

-3

to get

-15

.

4

Perform the subtraction.

8 + 15 = 23

Subtracting a negative number is mathematically identical to adding its positive counterpart.

Key Concept

Evaluating expressions using substitution and the order of operations.

Question 180Question →

A student is asked to simplify the rational algebraic expression shown below.

\frac{(3a^2b^{-3})^2}{3a^{-1}b^{-4}}

What is the fully simplified form of this expression with positive exponents?

\frac{3a^5}{b^2}

\frac{a^5}{b^2}

\frac{3a^3}{b^{10}}

a^6b^2

Show answer & explanation

Answer: $\frac{3a^5}{b^2}$

Answer

\frac{3a^5}{b^2}

To simplify the expression, first simplify the numerator by squaring every factor inside the parentheses:

(3)^2 \cdot (a^2)^2 \cdot (b^{-3})^2

, which yields

9a^4b^{-6}

. Next, divide this result by the denominator

3a^{-1}b^{-4}

. Divide the coefficients:

9 \div 3 = 3

. Then, apply the quotient rule by subtracting the exponents of like bases: for

a

,

4 - (-1) = 5

; for

b

,

-6 - (-4) = -2

. This gives

3a^5b^{-2}

. Finally, rewrite the expression with positive exponents by moving

b^{-2}

to the denominator, resulting in

\frac{3a^5}{b^2}

.

Step-by-Step Solution

1

Apply the power of a product and power of a power rules to the numerator:

(3a^2b^{-3})^2

.

9a^4b^{-6}

The exponent outside the parentheses applies to the coefficient and is multiplied by each variable's exponent.

2

Substitute the expanded numerator back into the fraction.

\frac{9a^4b^{-6}}{3a^{-1}b^{-4}}

This sets up the expression for the quotient rule.

3

Divide the numerical coefficients.

9 \div 3 = 3

Coefficients are divided normally.

4

Apply the quotient rule by subtracting the denominator's exponents from the numerator's exponents for bases

a

and

b

.

a^{4 - (-1)} = a^5

and

b^{-6 - (-4)} = b^{-2}

When dividing like bases, exponents are subtracted. Care must be taken with double negative signs.

5

Combine the results and convert any negative exponents to positive exponents by moving them to the denominator.

3a^5b^{-2} = \frac{3a^5}{b^2}

Standard mathematical notation requires final answers to be expressed using positive exponents.

Key Concept

Applying exponent rules (power, product, and quotient) with numerical coefficients and negative integers.

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