Question

question
simplify the following expression to simplest form using only positive exponents.
\\((64x^{-15}y^{24})^{\frac{2}{3}}\\)

Explanation:

Step1: Apply exponent rule \((ab)^n = a^n b^n\)

We can distribute the exponent \(\frac{2}{3}\) to each factor inside the parentheses:
\((64x^{-15}y^{24})^{\frac{2}{3}} = 64^{\frac{2}{3}} \cdot (x^{-15})^{\frac{2}{3}} \cdot (y^{24})^{\frac{2}{3}}\)

Step2: Simplify \(64^{\frac{2}{3}}\)

Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\). For \(64^{\frac{2}{3}}\), we know that \(64 = 4^3=2^6\), so:
\(64^{\frac{2}{3}}=(4^3)^{\frac{2}{3}} = 4^{3\times\frac{2}{3}} = 4^2 = 16\)

Step3: Simplify \((x^{-15})^{\frac{2}{3}}\)

Using the exponent rule \((a^m)^n=a^{m\times n}\), we have:
\((x^{-15})^{\frac{2}{3}}=x^{-15\times\frac{2}{3}}=x^{-10}\)
Since we need positive exponents, we use the rule \(a^{-n}=\frac{1}{a^n}\), so \(x^{-10}=\frac{1}{x^{10}}\)

Step4: Simplify \((y^{24})^{\frac{2}{3}}\)

Using the exponent rule \((a^m)^n = a^{m\times n}\):
\((y^{24})^{\frac{2}{3}}=y^{24\times\frac{2}{3}} = y^{16}\)

Step5: Combine the results

Multiply the simplified factors together:
\(64^{\frac{2}{3}} \cdot (x^{-15})^{\frac{2}{3}} \cdot (y^{24})^{\frac{2}{3}}=16\times\frac{1}{x^{10}}\times y^{16}=\frac{16y^{16}}{x^{10}}\)

Answer:

\(\frac{16y^{16}}{x^{10}}\)