Simplify ((x^(2/3))^2)/((x^2)^(7/3))

$\frac{{(x^{\frac{2}{3}})}^{2}}{{(x^{2})}^{\frac{7}{3}}}$

Step 1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{\frac{2}{3} \cdot 2}}{{(x^{2})}^{\frac{7}{3}}}$

Multiply $\frac{2}{3} \cdot 2$ .

Combine $\frac{2}{3}$ and $2$ .

$\frac{x^{\frac{2 \cdot 2}{3}}}{{(x^{2})}^{\frac{7}{3}}}$

Multiply $2$ by $2$ .

$\frac{x^{\frac{4}{3}}}{{(x^{2})}^{\frac{7}{3}}}$

Step 2

Multiply the exponents in ${(x^{2})}^{\frac{7}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{\frac{4}{3}}}{x^{2 (\frac{7}{3})}}$

Multiply $2 (\frac{7}{3})$ .

Combine $2$ and $\frac{7}{3}$ .

$\frac{x^{\frac{4}{3}}}{x^{\frac{2 \cdot 7}{3}}}$

Multiply $2$ by $7$ .

$\frac{x^{\frac{4}{3}}}{x^{\frac{14}{3}}}$

Step 3

Move $x^{\frac{4}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{x^{\frac{14}{3}} x^{- \frac{4}{3}}}$

Step 4

Multiply $x^{\frac{14}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{\frac{14}{3} - \frac{4}{3}}}$

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{14 - 4}{3}}}$

Subtract $4$ from $14$ .

$\frac{1}{x^{\frac{10}{3}}}$