Simplify ((x^(1/3))^2)/((x^3)^(7/3))

$\frac{{(x^{\frac{1}{3}})}^{2}}{{(x^{3})}^{\frac{7}{3}}}$

Step 1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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

$\frac{x^{\frac{1}{3} \cdot 2}}{{(x^{3})}^{\frac{7}{3}}}$

Combine $\frac{1}{3}$ and $2$ .

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

Step 2

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Move $x^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{x^{7} x^{- \frac{2}{3}}}$

Step 4

Multiply $x^{7}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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

$\frac{1}{x^{7 - \frac{2}{3}}}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{x^{7 \cdot \frac{3}{3} - \frac{2}{3}}}$

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

$\frac{1}{x^{\frac{7 \cdot 3}{3} - \frac{2}{3}}}$

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{7 \cdot 3 - 2}{3}}}$

Simplify the numerator.

Multiply $7$ by $3$ .

$\frac{1}{x^{\frac{21 - 2}{3}}}$

Subtract $2$ from $21$ .

$\frac{1}{x^{\frac{19}{3}}}$