Question

simplify the following expression to simplest form using only positive exponents.
\\(\left(32x^{-25}y^{-5}\
ight)^{\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 the expression \((32x^{-25}y^{-5})^{\frac{1}{5}}\) to get \(32^{\frac{1}{5}}(x^{-25})^{\frac{1}{5}}(y^{-5})^{\frac{1}{5}}\).

Step2: Simplify \(32^{\frac{1}{5}}\)

We know that \(32 = 2^5\), so \(32^{\frac{1}{5}}=(2^5)^{\frac{1}{5}}\). Using the power of a power rule \((a^m)^n = a^{mn}\), we have \((2^5)^{\frac{1}{5}} = 2^{5\times\frac{1}{5}} = 2^1 = 2\).

Step3: Simplify \((x^{-25})^{\frac{1}{5}}\)

Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(x^{-25\times\frac{1}{5}} = x^{-5}\). Since we need positive exponents, we use the rule \(a^{-n}=\frac{1}{a^n}\), so \(x^{-5}=\frac{1}{x^5}\).

Step4: Simplify \((y^{-5})^{\frac{1}{5}}\)

Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(y^{-5\times\frac{1}{5}} = y^{-1}\). Using the rule \(a^{-n}=\frac{1}{a^n}\), we have \(y^{-1}=\frac{1}{y}\).

Step5: Combine the results

Now we combine the results from Step2, Step3, and Step4. So we have \(2\times\frac{1}{x^5}\times\frac{1}{y}=\frac{2}{x^5y}\).

Answer:

\(\frac{2}{x^{5}y}\)