Basic Math Examples

${(\sqrt[3]{5} \cdot \sqrt{5})}^{6}$

Step 1

Multiply $\sqrt[3]{5} \sqrt{5}$ .

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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 $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

${(5^{\frac{1}{3}} \sqrt{5})}^{6}$

Step 1.1.2

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{2}{6}}$ .

${(5^{\frac{2}{6}} \sqrt{5})}^{6}$

Step 1.1.3

Rewrite $5^{\frac{2}{6}}$ as $\sqrt[6]{5^{2}}$ .

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

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 1.1.5

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

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

Step 1.1.6

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

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

Step 1.2

Combine using the product rule for radicals.

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

Step 1.3

Multiply $5^{2}$ by $5^{3}$ 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.

${\sqrt[6]{5^{2 + 3}}}^{6}$

Step 1.3.2

Add $2$ and $3$ .

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

Step 2

Raise $5$ to the power of $5$ .

${\sqrt[6]{3125}}^{6}$

Step 3

Rewrite ${\sqrt[6]{3125}}^{6}$ as $3125$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{3125}$ as $3125^{\frac{1}{6}}$ .

${(3125^{\frac{1}{6}})}^{6}$

Step 3.2

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

$3125^{\frac{1}{6} \cdot 6}$

Step 3.3

Combine $\frac{1}{6}$ and $6$ .

$3125^{\frac{6}{6}}$

Step 3.4

Cancel the common factor of $6$ .

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Step 3.4.1

Cancel the common factor.

$3125^{\frac{6}{6}}$

Step 3.4.2

Rewrite the expression.

$3125^{1}$

Step 3.5

Evaluate the exponent.

$3125$