Algebra Examples

$\sqrt[4]{4} \cdot \sqrt[3]{4}$

Step 1

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

$\sqrt[4]{2^{2}} \cdot \sqrt[3]{4}$

Step 2

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

$\sqrt{\sqrt{2^{2}}} \cdot \sqrt[3]{4}$

Step 3

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

$\sqrt{2} \cdot \sqrt[3]{4}$

Step 4

Multiply $\sqrt{2} \sqrt[3]{4}$ .

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

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

$2^{\frac{1}{2}} \sqrt[3]{4}$

Step 4.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

$2^{\frac{3}{6}} \sqrt[3]{4}$

Step 4.1.3

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$\sqrt[6]{2^{3}} \sqrt[3]{4}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\sqrt[6]{2^{3}} \cdot 4^{\frac{1}{3}}$

Step 4.1.5

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$\sqrt[6]{2^{3}} \cdot 4^{\frac{2}{6}}$

Step 4.1.6

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$\sqrt[6]{2^{3}} \sqrt[6]{4^{2}}$

Step 4.2

Combine using the product rule for radicals.

$\sqrt[6]{2^{3} \cdot 4^{2}}$

Step 4.3

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

$\sqrt[6]{2^{3} \cdot {(2^{2})}^{2}}$

Step 4.4

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

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Step 4.4.1

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

$\sqrt[6]{2^{3} \cdot 2^{2 \cdot 2}}$

Step 4.4.2

Multiply $2$ by $2$ .

$\sqrt[6]{2^{3} \cdot 2^{4}}$

Step 4.5

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

$\sqrt[6]{2^{3 + 4}}$

Step 4.6

Add $3$ and $4$ .

$\sqrt[6]{2^{7}}$

Step 5

Raise $2$ to the power of $7$ .

$\sqrt[6]{128}$

Step 6

Rewrite $128$ as $2^{6} \cdot 2$ .

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Step 6.1

Factor $64$ out of $128$ .

$\sqrt[6]{64 (2)}$

Step 6.2

Rewrite $64$ as $2^{6}$ .

$\sqrt[6]{2^{6} \cdot 2}$

Step 7

Pull terms out from under the radical.

$2 \sqrt[6]{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[6]{2}$

Decimal Form:

$2.24492409 \dots$