Precalculus Examples

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

Step 1

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

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Step 1.1

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

${\sqrt{5}}^{\sqrt{2} \cdot 2}$

Step 1.2

Move $2$ to the left of $\sqrt{2}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Divide $\sqrt{2}$ by $1$ .

$5^{\sqrt{2}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5^{\sqrt{2}}$

Decimal Form:

$9.73851774 \dots$