Question

simplify the following expression to simplest form using only positive exp
\\(\left(32x^{-25}y^{-40}\
ight)^{\frac{-3}{5}}\\)

Explanation:

Step1: Apply power of a product rule

We use the rule \((ab)^n = a^n b^n\) to the expression \((32x^{-25}y^{-40})^{-\frac{3}{5}}\). So we get \(32^{-\frac{3}{5}}(x^{-25})^{-\frac{3}{5}}(y^{-40})^{-\frac{3}{5}}\).

Step2: Simplify the constant term

First, recall that \(32 = 2^5\). So \(32^{-\frac{3}{5}}=(2^5)^{-\frac{3}{5}}\). Using the power of a power rule \((a^m)^n=a^{mn}\), we have \((2^5)^{-\frac{3}{5}} = 2^{5\times(-\frac{3}{5})}=2^{-3}\). And \(2^{-3}=\frac{1}{2^3}=\frac{1}{8}\).

Step3: Simplify the \(x\)-term

For \((x^{-25})^{-\frac{3}{5}}\), using the power of a power rule \((a^m)^n = a^{mn}\), we get \(x^{-25\times(-\frac{3}{5})}=x^{15}\) (since \(- 25\times(-\frac{3}{5})=15\)).

Step4: Simplify the \(y\)-term

For \((y^{-40})^{-\frac{3}{5}}\), using the power of a power rule \((a^m)^n=a^{mn}\), we get \(y^{-40\times(-\frac{3}{5})}=y^{24}\) (since \(-40\times(-\frac{3}{5}) = 24\)).

Step5: Combine the terms

Now we combine the simplified constant, \(x\)-term, and \(y\)-term. We have \(\frac{1}{8}x^{15}y^{24}\).

Answer:

\(\frac{1}{8}x^{15}y^{24}\)