If w and z are positive real numbers and k is a constant such that the expression \frac{(w… — ACT

If $w$ and $z$ are positive real numbers and $k$ is a constant such that the expression $\frac{(w^{3/2} z^{-1})^4}{(w z^k)^2}$ is equivalent to $w^4 z^6$ , what is the value of $k$ ?

Answer: -5

Answer

The value of

k

-5

Applying the exponent rules, the expression simplifies to

w^4 z^{-4-2k}

. Equating the exponent of

z

to the exponent in the target expression

w^4 z^6

gives

-4-2k = 6

, which solves to

k = -5

Step-by-Step Solution

Apply the power of a product rule to the expression in the numerator

(w^{3/2} z^{-1})^4 = w^6 z^{-4}

When raising a product to a power, multiply the exponent of each factor by the outer exponent:

(x^a y^b)^c = x^{ac} y^{bc}

Apply the power of a product rule to the expression in the denominator

(w z^k)^2 = w^2 z^{2k}

Multiply the exponent of each factor in the denominator by

2

Divide the numerator by the denominator using the quotient rule for exponents

\frac{w^6 z^{-4}}{w^2 z^{2k}} = w^4 z^{-4-2k}

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^a}{x^b} = x^{a-b}

Equate the exponent of

z

in the simplified expression to the exponent of

z

in the target expression

-4 - 2k = 6

Since the simplified expression is equivalent to

w^4 z^6

, the exponents of the corresponding variable bases must be equal.

Solve the linear equation for

k

k = -5

Add

4

to both sides of the equation to get

-2k = 10

, then divide both sides by

-2

Key Concept

Properties of Exponents in Algebraic Expressions

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