Question

question 2
rewrite the expression $x^{\frac{2}{3}} \cdot x^{\frac{5}{3}}$ to radical form
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Explanation:

Step1: Use exponent rule for multiplication

When multiplying exponents with the same base, we add the exponents: \(a^m \cdot a^n = a^{m + n}\). So for \(x^{\frac{2}{3}} \cdot x^{\frac{5}{3}}\), we add \(\frac{2}{3}\) and \(\frac{5}{3}\).
\(x^{\frac{2}{3}+\frac{5}{3}} = x^{\frac{7}{3}}\)

Step2: Convert exponent to radical form

The exponent form \(a^{\frac{m}{n}}\) can be written as \(\sqrt[n]{a^m}\). For \(x^{\frac{7}{3}}\), \(n = 3\) and \(m = 7\), so it becomes \(\sqrt[3]{x^7}\) (or we can also simplify \(x^7\) as \(x^6 \cdot x\), so \(\sqrt[3]{x^6 \cdot x}=x^2\sqrt[3]{x}\), but the question just asks for radical form, so \(\sqrt[3]{x^7}\) or \(x^2\sqrt[3]{x}\) are both correct. Here we'll use \(\sqrt[3]{x^7}\) or simplify further: \(x^2\sqrt[3]{x}\) since \(x^7 = x^{6 + 1}=x^6 \cdot x=(x^2)^3 \cdot x\), so \(\sqrt[3]{(x^2)^3 \cdot x}=x^2\sqrt[3]{x}\)).

Answer:

\(x^2\sqrt[3]{x}\) (or \(\sqrt[3]{x^7}\))