Question

simplify the following expression to simplest form using only positive exponents.
\\((32x^{-30}y^{20})^\frac{1}{5}\\)

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So we can apply this to \((32x^{-30}y^{20})^{\frac{1}{5}}\) as follows:
\(32^{\frac{1}{5}} \cdot (x^{-30})^{\frac{1}{5}} \cdot (y^{20})^{\frac{1}{5}}\)

Step2: Simplify each term

• For \(32^{\frac{1}{5}}\): We know that \(2^5 = 32\), so \(32^{\frac{1}{5}} = 2\).
• For \((x^{-30})^{\frac{1}{5}}\): Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(x^{-30 \cdot \frac{1}{5}} = x^{-6}\).
• For \((y^{20})^{\frac{1}{5}}\): Using the power of a power rule, we get \(y^{20 \cdot \frac{1}{5}} = y^{4}\).

Step3: Convert negative exponents to positive

Recall that \(a^{-n} = \frac{1}{a^n}\), so \(x^{-6} = \frac{1}{x^6}\). Now we combine the terms:
\(2 \cdot \frac{1}{x^6} \cdot y^4 = \frac{2y^4}{x^6}\)

Answer:

\(\frac{2y^{4}}{x^{6}}\)