Basic Math Examples

${(- \sqrt[10]{32} \cdot \sqrt[4]{4})}^{2}$

Step 1

Rewrite $32$ as $2^{5}$ .

${(- \sqrt[10]{2^{5}} \cdot \sqrt[4]{4})}^{2}$

Step 2

Rewrite $\sqrt[10]{2^{5}}$ as $\sqrt{\sqrt[5]{2^{5}}}$ .

${(- \sqrt{\sqrt[5]{2^{5}}} \cdot \sqrt[4]{4})}^{2}$

Step 3

Pull terms out from under the radical, assuming real numbers.

${(- \sqrt{2} \cdot \sqrt[4]{4})}^{2}$

Step 4

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

${(- \sqrt{2} \cdot \sqrt[4]{2^{2}})}^{2}$

Step 5

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

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

Step 6

Pull terms out from under the radical, assuming positive real numbers.

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

Step 7

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 7.1

Raise $\sqrt{2}$ to the power of $1$ .

${(- ({\sqrt{2}}^{1} \sqrt{2}))}^{2}$

Step 7.2

Raise $\sqrt{2}$ to the power of $1$ .

${(- ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))}^{2}$

Step 7.3

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

${(- {\sqrt{2}}^{1 + 1})}^{2}$

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

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

${(- {(2^{\frac{1}{2}})}^{2})}^{2}$

Step 8.2

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

${(- 2^{\frac{1}{2} \cdot 2})}^{2}$

Step 8.3

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

${(- 2^{\frac{2}{2}})}^{2}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

${(- 2^{\frac{2}{2}})}^{2}$

Step 8.4.2

Rewrite the expression.

${(- 2^{1})}^{2}$

Step 8.5

Evaluate the exponent.

${(- 1 \cdot 2)}^{2}$

Step 9

Simplify the expression.

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Step 9.1

Multiply $- 1$ by $2$ .

${(- 2)}^{2}$

Step 9.2

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

$4$