Algebra Examples

$\frac{{(3 x^{\frac{2}{3}})}^{3}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $3 x^{\frac{2}{3}}$ .

$\frac{3^{3} {(x^{\frac{2}{3}})}^{3}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 1.2

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

$\frac{27 {(x^{\frac{2}{3}})}^{3}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 1.3

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

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Step 1.3.1

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

$\frac{27 x^{\frac{2}{3} \cdot 3}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 1.3.2

Cancel the common factor of $3$ .

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Step 1.3.2.1

Cancel the common factor.

$\frac{27 x^{\frac{2}{3} \cdot 3}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 1.3.2.2

Rewrite the expression.

$\frac{27 x^{2}}{{(6 x^{- \frac{1}{3}})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $6 x^{- \frac{1}{3}}$ .

$\frac{27 x^{2}}{6^{2} {(x^{- \frac{1}{3}})}^{2}}$

Step 2.2

Raise $6$ to the power of $2$ .

$\frac{27 x^{2}}{36 {(x^{- \frac{1}{3}})}^{2}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{27 x^{2}}{36 x^{- \frac{1}{3} \cdot 2}}$

Step 2.3.2

Multiply $- \frac{1}{3} \cdot 2$ .

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Step 2.3.2.1

Multiply $2$ by $- 1$ .

$\frac{27 x^{2}}{36 x^{- 2 (\frac{1}{3})}}$

Step 2.3.2.2

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

$\frac{27 x^{2}}{36 x^{\frac{- 2}{3}}}$

Step 2.3.3

Move the negative in front of the fraction.

$\frac{27 x^{2}}{36 x^{- \frac{2}{3}}}$

Step 2.4

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

$\frac{27 x^{2}}{36 \frac{1}{x^{\frac{2}{3}}}}$

Step 3

Combine $36$ and $\frac{1}{x^{\frac{2}{3}}}$ .

$\frac{27 x^{2}}{\frac{36}{x^{\frac{2}{3}}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$27 x^{2} \frac{x^{\frac{2}{3}}}{36}$

Step 5

Cancel the common factor of $9$ .

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Step 5.1

Factor $9$ out of $27 x^{2}$ .

$9 (3 x^{2}) \frac{x^{\frac{2}{3}}}{36}$

Step 5.2

Factor $9$ out of $36$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

Combine $3$ and $\frac{x^{\frac{2}{3}}}{4}$ .

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

Step 7

Combine $x^{2}$ and $\frac{3 x^{\frac{2}{3}}}{4}$ .

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

Step 8

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

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Step 8.1

Move $x^{\frac{2}{3}}$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Combine the numerators over the common denominator.

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

Step 8.6

Simplify the numerator.

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Step 8.6.1

Multiply $2$ by $3$ .

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

Step 8.6.2

Add $2$ and $6$ .

$\frac{x^{\frac{8}{3}} \cdot 3}{4}$

Step 9

Move $3$ to the left of $x^{\frac{8}{3}}$ .

$\frac{3 x^{\frac{8}{3}}}{4}$