For any positive real number y, the expression \frac{(y^3)^{1/2}}{y^{2/3}} is equivalent t… — ACT

For any positive real number $y$ , the expression $\frac{(y^3)^{1/2}}{y^{2/3}}$ is equivalent to which of the following?

The simplified expression is

y^{5/6}

Applying the power of a power rule to the numerator gives

(y^3)^{1/2} = y^{3/2}

. Then, applying the quotient rule to divide by

y^{2/3}

requires subtracting the exponents:

\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}

. This results in the equivalent expression

y^{5/6}

Apply the power of a power rule to the numerator

(y^3)^{1/2}

y^{3 \cdot 1/2} = y^{3/2}

When raising a power to a power, multiply the exponents.

Apply the quotient of powers rule to divide

y^{3/2}

y^{2/3}

y^{3/2 - 2/3}

When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Subtract the fractions in the exponent by finding a common denominator.

y^{9/6 - 4/6} = y^{5/6}

A common denominator for 2 and 3 is 6. Rewrite the fractions and subtract their numerators.

Properties of Exponents in Algebraic Expressions

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