Algebra Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.2

Combine using the product rule for radicals.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Add $3$ and $2$ .

${\sqrt[6]{x^{5}}}^{6}$

Step 2

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

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{5}}$ as $x^{\frac{5}{6}}$ .

${(x^{\frac{5}{6}})}^{6}$

Step 2.2

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

$x^{\frac{5}{6} \cdot 6}$

Step 2.3

Combine $\frac{5}{6}$ and $6$ .

$x^{\frac{5 \cdot 6}{6}}$

Step 2.4

Multiply $5$ by $6$ .

$x^{\frac{30}{6}}$

Step 2.5

Cancel the common factor of $30$ and $6$ .

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Step 2.5.1

Factor $6$ out of $30$ .

$x^{\frac{6 \cdot 5}{6}}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $6$ out of $6$ .

$x^{\frac{6 \cdot 5}{6 (1)}}$

Step 2.5.2.2

Cancel the common factor.

$x^{\frac{6 \cdot 5}{6 \cdot 1}}$

Step 2.5.2.3

Rewrite the expression.

$x^{\frac{5}{1}}$

Step 2.5.2.4

Divide $5$ by $1$ .

$x^{5}$