Algebra Examples

$\sqrt[4]{25} \cdot \sqrt[3]{5}$

Step 1

Rewrite $25$ as $5^{2}$ .

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

Step 2

Rewrite $\sqrt[4]{5^{2}}$ as $\sqrt{\sqrt{5^{2}}}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

Multiply $5^{3}$ by $5^{2}$ by adding the exponents.

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Step 5.3.1

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

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

Step 5.3.2

Add $3$ and $2$ .

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

Step 6

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

$\sqrt[6]{3125}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{3125}$

Decimal Form:

$3.82362245 \dots$