Precalculus Examples

$\sqrt{\sqrt{\sqrt{x}}} = 2$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\sqrt{\sqrt{x}}}}^{2} = 2^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({\sqrt{\sqrt{x}}}^{\frac{1}{2}})}^{2} = 2^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({\sqrt{\sqrt{x}}}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({\sqrt{\sqrt{x}}}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${\sqrt{\sqrt{x}}}^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${\sqrt{\sqrt{x}}}^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${\sqrt{\sqrt{x}}}^{1} = 2^{2}$

Step 2.2.1.2

Rewrite $\sqrt{\sqrt{x}}$ as $\sqrt[4]{x}$ .

${\sqrt[4]{x}}^{1} = 2^{2}$

Step 2.2.1.3

Simplify.

$\sqrt[4]{x} = 2^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $2$ to the power of $2$ .

$\sqrt[4]{x} = 4$

Step 3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x}}^{4} = 4^{4}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${(x^{\frac{1}{4}})}^{4} = 4^{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

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Step 4.2.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 4.2.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = 4^{4}$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 4^{4}$

Step 4.2.1.1.2.2

Rewrite the expression.

$x^{1} = 4^{4}$

Step 4.2.1.2

Simplify.

$x = 4^{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $4$ to the power of $4$ .

$x = 256$