Simplify the following expression to simplest form using only positive exponents: (16x^{20}y^{-8})^{\frac{3}{4}}

Steps to Solve

1. Apply the power rule to each term inside the parentheses.

The power rule states that $(ab)^n = a^n b^n$. Apply this to the given expression: $(16x^{20}y^{-8})^{\frac{3}{4}} = 16^{\frac{3}{4}} (x^{20})^{\frac{3}{4}} (y^{-8})^{\frac{3}{4}}$

2. Simplify $16^{\frac{3}{4}}$.

$16^{\frac{3}{4}}$ can be rewritten as $(16^{\frac{1}{4}})^3$. Since $16^{\frac{1}{4}} = \sqrt[4]{16} = 2$, we have $(2)^3 = 8$.

3. Simplify $(x^{20})^{\frac{3}{4}}$.

Using the power rule $(a^m)^n = a^{mn}$, we get $x^{20 \cdot \frac{3}{4}} = x^{\frac{60}{4}} = x^{15}$.

4. Simplify $(y^{-8})^{\frac{3}{4}}$.

Again using the power rule $(a^m)^n = a^{mn}$, we get $y^{-8 \cdot \frac{3}{4}} = y^{\frac{-24}{4}} = y^{-6}$.

5. Rewrite the expression with positive exponents.

We now have $8x^{15}y^{-6}$. To express with positive exponents only, we rewrite $y^{-6}$ as $\frac{1}{y^6}$.

Therefore, the simplified expression is $\frac{8x^{15}}{y^6}$.