Question

simplify the following expression to simplest form using only positive exponents. \\(\left(32x^{-35}y^{-20}\
ight)^{\frac{-3}{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 each factor inside the parentheses:
\((32x^{-35}y^{-20})^{-\frac{3}{5}} = 32^{-\frac{3}{5}} \cdot (x^{-35})^{-\frac{3}{5}} \cdot (y^{-20})^{-\frac{3}{5}}\)

Step2: Simplify the constant term

We know that \(32 = 2^5\), so substitute that in:
\(32^{-\frac{3}{5}}=(2^5)^{-\frac{3}{5}}\)
Using the power of a power rule \((a^m)^n = a^{mn}\), we get:
\((2^5)^{-\frac{3}{5}} = 2^{5\times(-\frac{3}{5})}=2^{-3}\)
And \(2^{-3}=\frac{1}{2^3}=\frac{1}{8}\)

Step3: Simplify the \(x\)-term

For \((x^{-35})^{-\frac{3}{5}}\), use the power of a power rule:
\((x^{-35})^{-\frac{3}{5}}=x^{-35\times(-\frac{3}{5})}=x^{21}\)

Step4: Simplify the \(y\)-term

For \((y^{-20})^{-\frac{3}{5}}\), use the power of a power rule:
\((y^{-20})^{-\frac{3}{5}}=y^{-20\times(-\frac{3}{5})}=y^{12}\)

Step5: Combine the results

Now we combine the simplified constant, \(x\)-term, and \(y\)-term:
\(32^{-\frac{3}{5}} \cdot (x^{-35})^{-\frac{3}{5}} \cdot (y^{-20})^{-\frac{3}{5}}=\frac{1}{8}x^{21}y^{12}\)

Answer:

\(\frac{1}{8}x^{21}y^{12}\)