Question

59 simplify \\(\frac{x^{\frac{7}{3}}}{x^{\frac{2}{3}}}\\). write your answer in radical form.

Explanation:

Step1: Apply exponent rule for division

When dividing exponents with the same base, subtract the exponents: \( \frac{x^a}{x^b} = x^{a - b} \). Here, \( a=\frac{7}{3} \) and \( b = \frac{2}{3} \), so \( \frac{x^{\frac{7}{3}}}{x^{\frac{2}{3}}}=x^{\frac{7}{3}-\frac{2}{3}} \).

Step2: Subtract the exponents

Calculate \( \frac{7}{3}-\frac{2}{3}=\frac{7 - 2}{3}=\frac{5}{3} \). So we have \( x^{\frac{5}{3}} \).

Step3: Convert to radical form

Recall that \( x^{\frac{m}{n}}=\sqrt[n]{x^m} \). For \( x^{\frac{5}{3}} \), \( n = 3 \) and \( m = 5 \), so \( x^{\frac{5}{3}}=\sqrt[3]{x^5} \). We can also simplify \( x^5 \) as \( x^3\cdot x^2 \), so \( \sqrt[3]{x^5}=\sqrt[3]{x^3\cdot x^2}=x\sqrt[3]{x^2} \) (since \( \sqrt[3]{x^3}=x \)).

Answer:

\( x\sqrt[3]{x^2} \) (or \( \sqrt[3]{x^5} \))