For all positive real numbers x and y, the expression \frac{(3x^2 y^{-3})^3 \cdot (2x^{-1}… — ACT

For all positive real numbers

x

and

y

, the expression

\frac{(3x^2 y^{-3})^3 \cdot (2x^{-1} y^2)^2}{(6x^3 y^{-2})^2}

can be simplified to the form

A x^a y^b

, where

A

a

, and

b

are integers. What is the value of the sum

A + a + b

Answer: 0

The value of the sum

A + a + b

is 0.

By applying the rules of exponents systematically, the expression simplifies to

3 x^{-2} y^{-1}

. Comparing this to

A x^a y^b

yields

A = 3

a = -2

, and

b = -1

. The sum is

3 + (-2) + (-1) = 0

Simplify the first term in the numerator

27x^6y^{-9}

Apply the power of a product rule and power of a power rule to

(3x^2 y^{-3})^3

Simplify the second term in the numerator

4x^{-2}y^4

Apply the power of a product rule and power of a power rule to

(2x^{-1} y^2)^2

Multiply the simplified terms in the numerator together

108x^4y^{-5}

Multiply coefficients and add the exponents of like bases.

Simplify the denominator

36x^6y^{-4}

Apply the power of a product rule and power of a power rule to

(6x^3 y^{-2})^2

Divide the numerator by the denominator

3x^{-2}y^{-1}

Divide the coefficients and subtract the denominator exponents from the numerator exponents for like bases.

Sum the constants

A

a

, and

b

Identify

A = 3

a = -2

b = -1

from the expression

3x^{-2}y^{-1}

, and calculate

3 + (-2) + (-1) = 0

Properties of Exponents in Algebraic Expressions

Question