Simplify the Radical Expression fourth root of 1296/25

$\sqrt[4]{\frac{1296}{25}}$

Step 1

Rewrite $1296$ as $36^{2}$ .

$\sqrt[4]{\frac{36^{2}}{25}}$

Step 2

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

$\sqrt[4]{\frac{36^{2}}{5^{2}}}$

Step 3

Rewrite $\frac{36^{2}}{5^{2}}$ as ${(\frac{36}{5})}^{2}$ .

$\sqrt[4]{{(\frac{36}{5})}^{2}}$

Step 4

Rewrite $\sqrt[4]{{(\frac{36}{5})}^{2}}$ as $\sqrt{\sqrt{{(\frac{36}{5})}^{2}}}$ .

$\sqrt{\sqrt{{(\frac{36}{5})}^{2}}}$

Step 5

Pull terms out from under the radical.

$\sqrt{| \frac{36}{5} |}$

Step 6

$\frac{36}{5}$ is approximately $7.2$ which is positive so remove the absolute value

$\sqrt{\frac{36}{5}}$

Step 7

Rewrite $\sqrt{\frac{36}{5}}$ as $\frac{\sqrt{36}}{\sqrt{5}}$ .

$\frac{\sqrt{36}}{\sqrt{5}}$

Step 8

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

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

Pull terms out from under the radical.

$\frac{| 6 |}{\sqrt{5}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{6}{\sqrt{5}}$

Step 9

Multiply $\frac{6}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{6}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 10

Combine and simplify the denominator.

Multiply $\frac{6}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{6 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{6 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{6 \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$\frac{6 \sqrt{5}}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{6 \sqrt{5}}{5}$

Decimal Form:

$2.68328157 \dots$

Simplify the Radical Expression fourth root of 1296/25

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