Algebra Examples

${(\sqrt[3]{x^{4}} \cdot \sqrt{x^{5}})}^{- 2}$

Step 1

Factor out $x^{3}$ .

${(\sqrt[3]{x^{3} x} \cdot \sqrt{x^{5}})}^{- 2}$

Step 2

Pull terms out from under the radical.

${(x \sqrt[3]{x} \cdot \sqrt{x^{5}})}^{- 2}$

Step 3

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

Tap for more steps...

Step 3.1

Factor out $x^{4}$ .

${(x \sqrt[3]{x} \cdot \sqrt{x^{4} x})}^{- 2}$

Step 3.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x \sqrt[3]{x} \cdot \sqrt{{(x^{2})}^{2} x})}^{- 2}$

Step 4

Pull terms out from under the radical.

${(x \sqrt[3]{x} \cdot (x^{2} \sqrt{x}))}^{- 2}$

Step 5

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

Tap for more steps...

Step 5.1

Move $x^{2}$ .

${(x^{2} x \sqrt[3]{x} \cdot \sqrt{x})}^{- 2}$

Step 5.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 5.2.1

Raise $x$ to the power of $1$ .

${(x^{2} x^{1} \sqrt[3]{x} \cdot \sqrt{x})}^{- 2}$

Step 5.2.2

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

${(x^{2 + 1} \sqrt[3]{x} \cdot \sqrt{x})}^{- 2}$

Step 5.3

Add $2$ and $1$ .

${(x^{3} \sqrt[3]{x} \cdot \sqrt{x})}^{- 2}$

Step 6

Multiply $x^{3} \sqrt[3]{x} \sqrt{x}$ .

Tap for more steps...

Step 6.1

Rewrite the expression using the least common index of $6$ .

Tap for more steps...

Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{3} (x^{\frac{1}{2}} \sqrt[3]{x}))}^{- 2}$

Step 6.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

${(x^{3} (x^{\frac{3}{6}} \sqrt[3]{x}))}^{- 2}$

Step 6.1.3

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

${(x^{3} (\sqrt[6]{x^{3}} \sqrt[3]{x}))}^{- 2}$

Step 6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 6.1.5

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

${(x^{3} (\sqrt[6]{x^{3}} x^{\frac{2}{6}}))}^{- 2}$

Step 6.1.6

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

${(x^{3} (\sqrt[6]{x^{3}} \sqrt[6]{x^{2}}))}^{- 2}$

Step 6.2

Combine using the product rule for radicals.

${(x^{3} \sqrt[6]{x^{3} x^{2}})}^{- 2}$

Step 6.3

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

Tap for more steps...

Step 6.3.1

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

${(x^{3} \sqrt[6]{x^{3 + 2}})}^{- 2}$

Step 6.3.2

Add $3$ and $2$ .

${(x^{3} \sqrt[6]{x^{5}})}^{- 2}$

Step 7

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

$\frac{1}{{(x^{3} \sqrt[6]{x^{5}})}^{2}}$

Step 8

Simplify the denominator.

Tap for more steps...

Step 8.1

Apply the product rule to $x^{3} \sqrt[6]{x^{5}}$ .

$\frac{1}{{(x^{3})}^{2} {\sqrt[6]{x^{5}}}^{2}}$

Step 8.2

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

Tap for more steps...

Step 8.2.1

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

$\frac{1}{x^{3 \cdot 2} {\sqrt[6]{x^{5}}}^{2}}$

Step 8.2.2

Multiply $3$ by $2$ .

$\frac{1}{x^{6} {\sqrt[6]{x^{5}}}^{2}}$

Step 8.3

Rewrite ${\sqrt[6]{x^{5}}}^{2}$ as $\sqrt[6]{{(x^{5})}^{2}}$ .

$\frac{1}{x^{6} \sqrt[6]{{(x^{5})}^{2}}}$

Step 8.4

Multiply the exponents in ${(x^{5})}^{2}$ .

Tap for more steps...

Step 8.4.1

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

$\frac{1}{x^{6} \sqrt[6]{x^{5 \cdot 2}}}$

Step 8.4.2

Multiply $5$ by $2$ .

$\frac{1}{x^{6} \sqrt[6]{x^{10}}}$

Step 8.5

Factor out $x^{6}$ .

$\frac{1}{x^{6} \sqrt[6]{x^{6} x^{4}}}$

Step 8.6

Pull terms out from under the radical.

$\frac{1}{x^{6} (x \sqrt[6]{x^{4}})}$

Step 8.7

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{1}{x^{6} (x \sqrt[6]{{(x^{2})}^{2}})}$

Step 8.8

Rewrite $\sqrt[6]{{(x^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2})}^{2}}}$ .

$\frac{1}{x^{6} (x \sqrt[3]{\sqrt{{(x^{2})}^{2}}})}$

Step 8.9

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{x^{6} x \sqrt[3]{x^{2}}}$

Step 8.10

Multiply $x^{6}$ by $x$ by adding the exponents.

Tap for more steps...

Step 8.10.1

Multiply $x^{6}$ by $x$ .

Tap for more steps...

Step 8.10.1.1

Raise $x$ to the power of $1$ .

$\frac{1}{x^{6} x^{1} \sqrt[3]{x^{2}}}$

Step 8.10.1.2

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

$\frac{1}{x^{6 + 1} \sqrt[3]{x^{2}}}$

Step 8.10.2

Add $6$ and $1$ .

$\frac{1}{x^{7} \sqrt[3]{x^{2}}}$

Step 9

Multiply $\frac{1}{x^{7} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\frac{1}{x^{7} \sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 10

Combine and simplify the denominator.

Tap for more steps...

Step 10.1

Multiply $\frac{1}{x^{7} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} \sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}}$

Step 10.2

Move $\sqrt[3]{x^{2}}$ .

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

Step 10.3

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

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

Step 10.4

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

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} {\sqrt[3]{x^{2}}}^{1 + 2}}$

Step 10.5

Add $1$ and $2$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} {\sqrt[3]{x^{2}}}^{3}}$

Step 10.6

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

Tap for more steps...

Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 10.6.2

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

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{2}{3} \cdot 3}}$

Step 10.6.3

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

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{2 \cdot 3}{3}}}$

Step 10.6.4

Multiply $2$ by $3$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{6}{3}}}$

Step 10.6.5

Cancel the common factor of $6$ and $3$ .

Tap for more steps...

Step 10.6.5.1

Factor $3$ out of $6$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{3 \cdot 2}{3}}}$

Step 10.6.5.2

Cancel the common factors.

Tap for more steps...

Step 10.6.5.2.1

Factor $3$ out of $3$ .

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

Step 10.6.5.2.2

Cancel the common factor.

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 10.6.5.2.3

Rewrite the expression.

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{\frac{2}{1}}}$

Step 10.6.5.2.4

Divide $2$ by $1$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7} x^{2}}$

Step 11

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 11.1

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

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{7 + 2}}$

Step 11.2

Add $7$ and $2$ .

$\frac{{\sqrt[3]{x^{2}}}^{2}}{x^{9}}$

Step 12

Simplify the numerator.

Tap for more steps...

Step 12.1

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as $\sqrt[3]{{(x^{2})}^{2}}$ .

$\frac{\sqrt[3]{{(x^{2})}^{2}}}{x^{9}}$

Step 12.2

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

Tap for more steps...

Step 12.2.1

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

$\frac{\sqrt[3]{x^{2 \cdot 2}}}{x^{9}}$

Step 12.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{x^{4}}}{x^{9}}$

Step 12.3

Factor out $x^{3}$ .

$\frac{\sqrt[3]{x^{3} x}}{x^{9}}$

Step 12.4

Pull terms out from under the radical.

$\frac{x \sqrt[3]{x}}{x^{9}}$

Step 13

Cancel the common factor of $x$ and $x^{9}$ .

Tap for more steps...

Step 13.1

Factor $x$ out of $x \sqrt[3]{x}$ .

$\frac{x (\sqrt[3]{x})}{x^{9}}$

Step 13.2

Cancel the common factors.

Tap for more steps...

Step 13.2.1

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

$\frac{x (\sqrt[3]{x})}{x \cdot x^{8}}$

Step 13.2.2

Cancel the common factor.

$\frac{x \sqrt[3]{x}}{x \cdot x^{8}}$

Step 13.2.3

Rewrite the expression.

$\frac{\sqrt[3]{x}}{x^{8}}$