Question

question
simplify the following expression to simplest form using only positive exponents.
$left(125x^{-18}y^{9}
ight)^{\frac{5}{3}}$

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 \((125x^{-18}y^{9})^{\frac{2}{3}}\) as follows:
\(125^{\frac{2}{3}}(x^{-18})^{\frac{2}{3}}(y^{9})^{\frac{2}{3}}\)

Step2: Simplify \(125^{\frac{2}{3}}\)

We know that \(125 = 5^3\), so \(125^{\frac{2}{3}}=(5^3)^{\frac{2}{3}}\). Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(5^{3\times\frac{2}{3}} = 5^2 = 25\).

Step3: Simplify \((x^{-18})^{\frac{2}{3}}\)

Using the power of a power rule \((a^m)^n = a^{mn}\), we have \(x^{-18\times\frac{2}{3}} = x^{-12}\). Since we need positive exponents, we use the rule \(a^{-n}=\frac{1}{a^n}\), so \(x^{-12}=\frac{1}{x^{12}}\).

Step4: Simplify \((y^{9})^{\frac{2}{3}}\)

Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(y^{9\times\frac{2}{3}} = y^{6}\).

Step5: Combine the results

Now we combine the simplified terms from steps 2, 3, and 4:
\(25\times\frac{1}{x^{12}}\times y^{6}=\frac{25y^{6}}{x^{12}}\)

Answer:

\(\frac{25y^{6}}{x^{12}}\)