Algebra Examples

$\sqrt{5} \cdot \sqrt[3]{5} \cdot \sqrt[4]{5}$

Step 1

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

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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 $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5^{\frac{1}{2}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Step 1.1.2

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

$5^{\frac{3}{6}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Step 1.1.3

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

$\sqrt[6]{5^{3}} \sqrt[3]{5} \cdot \sqrt[4]{5}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\sqrt[6]{5^{3}} \cdot 5^{\frac{1}{3}} \cdot \sqrt[4]{5}$

Step 1.1.5

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

$\sqrt[6]{5^{3}} \cdot 5^{\frac{2}{6}} \cdot \sqrt[4]{5}$

Step 1.1.6

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

$\sqrt[6]{5^{3}} \sqrt[6]{5^{2}} \cdot \sqrt[4]{5}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[6]{5^{3} \cdot 5^{2}} \cdot \sqrt[4]{5}$

Step 1.3

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

$\sqrt[6]{5^{3 + 2}} \cdot \sqrt[4]{5}$

Step 1.3.2

Add $3$ and $2$ .

$\sqrt[6]{5^{5}} \cdot \sqrt[4]{5}$

Step 2

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

$\sqrt[6]{3125} \cdot \sqrt[4]{5}$

Step 3

Multiply $\sqrt[6]{3125} \sqrt[4]{5}$ .

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Step 3.1

Rewrite the expression using the least common index of $12$ .

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Step 3.1.1

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

$3125^{\frac{1}{6}} \sqrt[4]{5}$

Step 3.1.2

Rewrite $3125^{\frac{1}{6}}$ as $3125^{\frac{2}{12}}$ .

$3125^{\frac{2}{12}} \sqrt[4]{5}$

Step 3.1.3

Rewrite $3125^{\frac{2}{12}}$ as $\sqrt[12]{3125^{2}}$ .

$\sqrt[12]{3125^{2}} \sqrt[4]{5}$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$\sqrt[12]{3125^{2}} \cdot 5^{\frac{1}{4}}$

Step 3.1.5

Rewrite $5^{\frac{1}{4}}$ as $5^{\frac{3}{12}}$ .

$\sqrt[12]{3125^{2}} \cdot 5^{\frac{3}{12}}$

Step 3.1.6

Rewrite $5^{\frac{3}{12}}$ as $\sqrt[12]{5^{3}}$ .

$\sqrt[12]{3125^{2}} \sqrt[12]{5^{3}}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[12]{3125^{2} \cdot 5^{3}}$

Step 3.3

Rewrite $3125$ as $5^{5}$ .

$\sqrt[12]{{(5^{5})}^{2} \cdot 5^{3}}$

Step 3.4

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

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Step 3.4.1

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

$\sqrt[12]{5^{5 \cdot 2} \cdot 5^{3}}$

Step 3.4.2

Multiply $5$ by $2$ .

$\sqrt[12]{5^{10} \cdot 5^{3}}$

Step 3.5

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

$\sqrt[12]{5^{10 + 3}}$

Step 3.6

Add $10$ and $3$ .

$\sqrt[12]{5^{13}}$

Step 4

Raise $5$ to the power of $13$ .

$\sqrt[12]{1220703125}$

Step 5

Rewrite $1220703125$ as $5^{12} \cdot 5$ .

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Step 5.1

Factor $244140625$ out of $1220703125$ .

$\sqrt[12]{244140625 (5)}$

Step 5.2

Rewrite $244140625$ as $5^{12}$ .

$\sqrt[12]{5^{12} \cdot 5}$

Step 6

Pull terms out from under the radical.

$5 \sqrt[12]{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[12]{5}$

Decimal Form:

$5.71764918 \dots$