Algebra Examples

$\sqrt[3]{5^{2}} \cdot \sqrt[4]{5^{3}}$

Step 1

Raise $5$ to the power of $2$ .

$\sqrt[3]{25} \cdot \sqrt[4]{5^{3}}$

Step 2

Raise $5$ to the power of $3$ .

$\sqrt[3]{25} \cdot \sqrt[4]{125}$

Step 3

Multiply $\sqrt[3]{25} \sqrt[4]{125}$ .

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Step 3.1

Rewrite the expression using the least common index of $12$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$25^{\frac{1}{3}} \sqrt[4]{125}$

Step 3.1.2

Rewrite $25^{\frac{1}{3}}$ as $25^{\frac{4}{12}}$ .

$25^{\frac{4}{12}} \sqrt[4]{125}$

Step 3.1.3

Rewrite $25^{\frac{4}{12}}$ as $\sqrt[12]{25^{4}}$ .

$\sqrt[12]{25^{4}} \sqrt[4]{125}$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{125}$ as $125^{\frac{1}{4}}$ .

$\sqrt[12]{25^{4}} \cdot 125^{\frac{1}{4}}$

Step 3.1.5

Rewrite $125^{\frac{1}{4}}$ as $125^{\frac{3}{12}}$ .

$\sqrt[12]{25^{4}} \cdot 125^{\frac{3}{12}}$

Step 3.1.6

Rewrite $125^{\frac{3}{12}}$ as $\sqrt[12]{125^{3}}$ .

$\sqrt[12]{25^{4}} \sqrt[12]{125^{3}}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[12]{25^{4} \cdot 125^{3}}$

Step 3.3

Rewrite $25$ as $5^{2}$ .

$\sqrt[12]{{(5^{2})}^{4} \cdot 125^{3}}$

Step 3.4

Multiply the exponents in ${(5^{2})}^{4}$ .

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Step 3.4.1

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

$\sqrt[12]{5^{2 \cdot 4} \cdot 125^{3}}$

Step 3.4.2

Multiply $2$ by $4$ .

$\sqrt[12]{5^{8} \cdot 125^{3}}$

Step 3.5

Rewrite $125$ as $5^{3}$ .

$\sqrt[12]{5^{8} \cdot {(5^{3})}^{3}}$

Step 3.6

Multiply the exponents in ${(5^{3})}^{3}$ .

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Step 3.6.1

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

$\sqrt[12]{5^{8} \cdot 5^{3 \cdot 3}}$

Step 3.6.2

Multiply $3$ by $3$ .

$\sqrt[12]{5^{8} \cdot 5^{9}}$

Step 3.7

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

$\sqrt[12]{5^{8 + 9}}$

Step 3.8

Add $8$ and $9$ .

$\sqrt[12]{5^{17}}$

Step 4

Raise $5$ to the power of $17$ .

$\sqrt[12]{762939453125}$

Step 5

Rewrite $762939453125$ as $5^{12} \cdot 3125$ .

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Step 5.1

Factor $244140625$ out of $762939453125$ .

$\sqrt[12]{244140625 (3125)}$

Step 5.2

Rewrite $244140625$ as $5^{12}$ .

$\sqrt[12]{5^{12} \cdot 3125}$

Step 6

Pull terms out from under the radical.

$| 5 | \sqrt[12]{3125}$

Step 7

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$5 \sqrt[12]{3125}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[12]{3125}$

Decimal Form:

$9.77704257 \dots$