Questions: Simplify. [ left(frac4 y3 y^-2right)^3 ] Write your answer using only positive exponents.

Solution Steps

To simplify the given expression, we need to first simplify the fraction inside the parentheses by applying the rules of exponents. Specifically, we will use the property that \( a^{-n} = \frac{1}{a^n} \) to convert negative exponents to positive ones. Then, we will raise the simplified fraction to the power of 3 by applying the power of a quotient rule, which states that \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).

We start with the expression

\[ \left(\frac{4 y}{3 y^{-2}}\right)^{3} \]

First, we simplify the fraction inside the parentheses. The term \( y^{-2} \) can be rewritten as \( \frac{1}{y^2} \), so we have:

\[ \frac{4 y}{3 y^{-2}} = \frac{4 y}{3 \cdot \frac{1}{y^2}} = \frac{4 y^3}{3} \]

Next, we raise the simplified fraction to the power of 3:

\[ \left(\frac{4 y^3}{3}\right)^{3} = \frac{(4)^{3} (y^3)^{3}}{(3)^{3}} = \frac{64 y^9}{27} \]

Final Answer