For all positive real numbers w, the expression \left(\frac{w^4 \cdot \sqrt{w}}{w^{-1.5}}\… — ACT

For all positive real numbers $w$ , the expression $\left(\frac{w^4 \cdot \sqrt{w}}{w^{-1.5}}\right)^{2/3}$ is equivalent to $w^k$ , where $k$ is a constant. What is the value of $k$ ?

Answer: 4

Answer

The value of

k

4

Applying the exponent rules in sequence: first, rewrite the square root as a fractional exponent to get

w^{0.5}

. Next, multiply the terms in the numerator by adding their exponents:

4 + 0.5 = 4.5

. Then, divide the numerator by the denominator by subtracting the denominator's exponent from the numerator's exponent:

4.5 - (-1.5) = 6

. Finally, raise this result to the

2/3

power by multiplying the exponents:

6 \times (2/3) = 4

. This yields

w^4

, so the constant exponent is

4

Step-by-Step Solution

Convert the radical expression to an exponential expression.

\sqrt{w} = w^{0.5}

Converting all terms to base

w

with decimal or fractional exponents makes it easier to apply exponent properties.

Apply the product rule of exponents to the numerator.

w^4 \cdot w^{0.5} = w^{4 + 0.5} = w^{4.5}

When multiplying exponential terms with the same base, add their exponents:

w^a \cdot w^b = w^{a+b}

Apply the quotient rule of exponents to the fraction.

\frac{w^{4.5}}{w^{-1.5}} = w^{4.5 - (-1.5)} = w^6

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

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

Apply the power rule of exponents to the simplified term.

(w^6)^{2/3} = w^{6 \cdot \frac{2}{3}} = w^4

When raising a power to another power, multiply the exponents:

(w^a)^b = w^{a \cdot b}

Key Concept

Properties of Exponents in Algebraic Expressions

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