Simplify (9x^(1/2))/((27x^-2)^(2/3))

$\frac{9 x^{\frac{1}{2}}}{{(27 x^{- 2})}^{\frac{2}{3}}}$

Step 1

Simplify the denominator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{9 x^{\frac{1}{2}}}{{(27 \frac{1}{x^{2}})}^{\frac{2}{3}}}$

Combine $27$ and $\frac{1}{x^{2}}$ .

$\frac{9 x^{\frac{1}{2}}}{{(\frac{27}{x^{2}})}^{\frac{2}{3}}}$

Apply the product rule to $\frac{27}{x^{2}}$ .

$\frac{9 x^{\frac{1}{2}}}{\frac{27^{\frac{2}{3}}}{{(x^{2})}^{\frac{2}{3}}}}$

Simplify the numerator.

Rewrite $27$ as $3^{3}$ .

$\frac{9 x^{\frac{1}{2}}}{\frac{{(3^{3})}^{\frac{2}{3}}}{{(x^{2})}^{\frac{2}{3}}}}$

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

$\frac{9 x^{\frac{1}{2}}}{\frac{3^{3 (\frac{2}{3})}}{{(x^{2})}^{\frac{2}{3}}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{9 x^{\frac{1}{2}}}{\frac{3^{3 (\frac{2}{3})}}{{(x^{2})}^{\frac{2}{3}}}}$

Rewrite the expression.

$\frac{9 x^{\frac{1}{2}}}{\frac{3^{2}}{{(x^{2})}^{\frac{2}{3}}}}$

Raise $3$ to the power of $2$ .

$\frac{9 x^{\frac{1}{2}}}{\frac{9}{{(x^{2})}^{\frac{2}{3}}}}$

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

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

$\frac{9 x^{\frac{1}{2}}}{\frac{9}{x^{2 (\frac{2}{3})}}}$

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

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

$\frac{9 x^{\frac{1}{2}}}{\frac{9}{x^{\frac{2 \cdot 2}{3}}}}$

Multiply $2$ by $2$ .

$\frac{9 x^{\frac{1}{2}}}{\frac{9}{x^{\frac{4}{3}}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$9 x^{\frac{1}{2}} \frac{x^{\frac{4}{3}}}{9}$

Step 3

Cancel the common factor of $9$ .

Factor $9$ out of $9 x^{\frac{1}{2}}$ .

$9 (x^{\frac{1}{2}}) \frac{x^{\frac{4}{3}}}{9}$

Cancel the common factor.

$9 x^{\frac{1}{2}} \frac{x^{\frac{4}{3}}}{9}$

Rewrite the expression.

$x^{\frac{1}{2}} x^{\frac{4}{3}}$

Step 4

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{4}{3}}$ by adding the exponents.

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

$x^{\frac{1}{2} + \frac{4}{3}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3}}$

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3} \cdot \frac{2}{2}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Multiply $2$ by $3$ .

$x^{\frac{3}{6} + \frac{4}{3} \cdot \frac{2}{2}}$

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$x^{\frac{3}{6} + \frac{4 \cdot 2}{3 \cdot 2}}$

Multiply $3$ by $2$ .

$x^{\frac{3}{6} + \frac{4 \cdot 2}{6}}$

Combine the numerators over the common denominator.

$x^{\frac{3 + 4 \cdot 2}{6}}$

Simplify the numerator.

Multiply $4$ by $2$ .

$x^{\frac{3 + 8}{6}}$

Add $3$ and $8$ .

$x^{\frac{11}{6}}$

Simplify (9x^(1/2))/((27x^-2)^(2/3))

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