If the expression \frac{(x^2 y^3)^4}{(x^{-1} y^2)^3} is written in the equivalent form x^a… — ACT

If the expression $\frac{(x^2 y^3)^4}{(x^{-1} y^2)^3}$ is written in the equivalent form $x^a y^b$ , what is the value of $a - b$ ?

Answer: 5

Simplifying the numerator yields

x^8 y^{12}

and simplifying the denominator yields

x^{-3} y^6

. Dividing these expressions by subtracting the exponents of like bases results in

x^{8 - (-3)} y^{12 - 6} = x^{11} y^6

. Comparing this to the expression

x^a y^b

shows that

a = 11

and

b = 6

. The value of

a - b

11 - 6 = 5

Simplify the numerator of the expression.

x^8 y^{12}

Apply the power of a power and power of a product properties of exponents:

(x^2 y^3)^4 = x^{2 \cdot 4} y^{3 \cdot 4}

Simplify the denominator of the expression.

x^{-3} y^6

Apply the power of a power and power of a product properties of exponents:

(x^{-1} y^2)^3 = x^{-1 \cdot 3} y^{2 \cdot 3}

Simplify the quotient by dividing the simplified numerator by the simplified denominator.

x^{11} y^6

Use the quotient property of exponents,

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

, to subtract the exponents:

8 - (-3) = 11

and

12 - 6 = 6

Identify the values of

a

and

b

and compute the difference

a - b

By comparing

x^{11} y^6

to the target form

x^a y^b

, we find

a = 11

and

b = 6

. Subtracting

b

from

a

yields

11 - 6 = 5

Properties of exponents including power of a power, power of a product, and quotient rules.

Question