For all x > 0 and y > 0, the expression \frac{(x^2y^3)^a}{(xy^2)^2} is equivalent to x^6y^… — SAT

For all $x > 0$ and $y > 0$ , the expression $\frac{(x^2y^3)^a}{(xy^2)^2}$ is equivalent to $x^6y^8$ , where $a$ is a constant. What is the value of $a$ ?

Answer: 4

Answer

Applying the rules of exponents, the expression

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

simplifies to

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

. Setting this equal to the equivalent expression

x^6y^8

gives the system of equations

2a - 2 = 6

and

3a - 4 = 8

. Solving either equation yields

a = 4

Step-by-Step Solution

Apply the power rule of exponents to the numerator and denominator.

Numerator:

x^{2a}y^{3a}

, Denominator:

x^2y^4

To expand the parentheses by multiplying the outer exponent with the inner exponents.

Apply the quotient rule of exponents to divide the numerator by the denominator.

x^{2a-2}y^{3a-4}

To simplify the rational expression by subtracting the exponents in the denominator from the exponents in the numerator.

Equate the simplified exponent of

x

to the exponent of

x

in the given equivalent expression.

2a - 2 = 6

Equivalent expressions must have identical exponents for corresponding variable bases.

Solve the linear equation for

a

a = 4

To find the constant value that satisfies the equation.

Verify the value of

a

using the exponents of

y

3(4) - 4 = 8

, which is true.

To ensure consistency across both variable exponents.

Key Concept

Equivalent algebraic expressions involving exponent rules

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