Question

simplify the following expression to simplest form using only positive exponents.
(8x^{24}y^{-9})^{\frac{1}{3}}

Explanation:

Step1: Apply 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 \((8x^{24}y^{-9})^{\frac{1}{3}}\) as follows:
\(8^{\frac{1}{3}}(x^{24})^{\frac{1}{3}}(y^{-9})^{\frac{1}{3}}\)

Step2: Simplify each term

• For the constant term: We know that \(8 = 2^3\), so \(8^{\frac{1}{3}}=(2^3)^{\frac{1}{3}}\). Using the power of a power rule \((a^m)^n = a^{mn}\), we get \(2^{3\times\frac{1}{3}} = 2^1 = 2\).
• For the \(x\)-term: Using the power of a power rule on \((x^{24})^{\frac{1}{3}}\), we have \(x^{24\times\frac{1}{3}} = x^8\).
• For the \(y\)-term: Using the power of a power rule on \((y^{-9})^{\frac{1}{3}}\), we get \(y^{-9\times\frac{1}{3}} = y^{-3}\). But we need positive exponents, and we know that \(a^{-n}=\frac{1}{a^n}\), so \(y^{-3}=\frac{1}{y^3}\).

Step3: Combine the simplified terms

Putting it all together, we have \(2\times x^8\times\frac{1}{y^3}=\frac{2x^8}{y^3}\).

Answer:

\(\frac{2x^8}{y^3}\)