Algebra Examples

$\sqrt{\sqrt[4]{8^{10}}}$

Step 1

Raise $8$ to the power of $10$ .

$\sqrt{\sqrt[4]{1073741824}}$

Step 2

Rewrite $1073741824$ as $128^{4} \cdot 4$ .

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Step 2.1

Factor $268435456$ out of $1073741824$ .

$\sqrt{\sqrt[4]{268435456 (4)}}$

Step 2.2

Rewrite $268435456$ as $128^{4}$ .

$\sqrt{\sqrt[4]{128^{4} \cdot 4}}$

Step 3

Pull terms out from under the radical.

$\sqrt{| 128 | \sqrt[4]{4}}$

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $128$ is $128$ .

$\sqrt{128 \sqrt[4]{4}}$

Step 5

Rewrite $4$ as $2^{2}$ .

$\sqrt{128 \sqrt[4]{2^{2}}}$

Step 6

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

$\sqrt{128 \sqrt{\sqrt{2^{2}}}}$

Step 7

Pull terms out from under the radical.

$\sqrt{128 \sqrt{| 2 |}}$

Step 8

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\sqrt{128 \sqrt{2}}$

Step 9

Rewrite $128 \sqrt{2}$ as $8^{2} \cdot (2 \sqrt{2})$ .

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Step 9.1

Factor $64$ out of $128$ .

$\sqrt{64 (2) \sqrt{2}}$

Step 9.2

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2} \cdot 2 \sqrt{2}}$

Step 9.3

Add parentheses.

$\sqrt{8^{2} \cdot (2 \sqrt{2})}$

Step 10

Pull terms out from under the radical.

$| 8 | \sqrt{2 \sqrt{2}}$

Step 11

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$8 \sqrt{2 \sqrt{2}}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{2 \sqrt{2}}$

Decimal Form:

$13.45434264 \dots$