If x and y are non-zero real numbers such that the expression \frac{(x^2 y^a)^3}{x^5 y^{-2… — ACT

If $x$ and $y$ are non-zero real numbers such that the expression $\frac{(x^2 y^a)^3}{x^5 y^{-2}}$ is equivalent to $x^b y^{14}$ for some integers $a$ and $b$ , what is the value of $a + b$ ?

Answer: 5

Answer

The value of

a + b

5

Simplifying the expression

\frac{(x^2 y^a)^3}{x^5 y^{-2}}

using the exponent rules yields

x^{6-5} y^{3a-(-2)} = x^1 y^{3a+2}

. Equating this to

x^b y^{14}

shows that

b = 1

and

3a + 2 = 14

. Solving for

a

gives

a = 4

. Thus, the sum

a + b

4 + 1 = 5

Step-by-Step Solution

Simplify the numerator of the given expression.

x^6 y^{3a}

Applying the power of a product rule

(uv)^n = u^n v^n

and the power of a power rule

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

(x^2 y^a)^3

results in

(x^2)^3 (y^a)^3 = x^6 y^{3a}

Simplify the quotient by subtracting exponents with the same base.

x^1 y^{3a+2}

Using the quotient rule

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

, the base

x

term becomes

x^{6-5} = x^1

, and the base

y

term becomes

y^{3a - (-2)} = y^{3a+2}

Equate the exponents of like bases to find the values of

a

and

b

b = 1

and

a = 4

Comparing

x^1 y^{3a+2}

x^b y^{14}

gives

b = 1

and

3a + 2 = 14

. Solving

3a + 2 = 14

yields

3a = 12

, which simplifies to

a = 4

Sum the values of

a

and

b

5

Adding

a

and

b

yields

4 + 1 = 5

Key Concept

Applying properties of exponents, including the power of a product, power of a power, and quotient rules, to simplify algebraic expressions.

Estimated Time:1m 30s

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