Algebra Examples

$\sqrt[4]{10^{5}} \cdot \sqrt[3]{10}$

Step 1

Raise $10$ to the power of $5$ .

$\sqrt[4]{100000} \cdot \sqrt[3]{10}$

Step 2

Rewrite $100000$ as $10^{4} \cdot 10$ .

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Step 2.1

Factor $10000$ out of $100000$ .

$\sqrt[4]{10000 (10)} \cdot \sqrt[3]{10}$

Step 2.2

Rewrite $10000$ as $10^{4}$ .

$\sqrt[4]{10^{4} \cdot 10} \cdot \sqrt[3]{10}$

Step 3

Pull terms out from under the radical.

$| 10 | \sqrt[4]{10} \cdot \sqrt[3]{10}$

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$10 \sqrt[4]{10} \cdot \sqrt[3]{10}$

Step 5

Multiply $10 \sqrt[4]{10} \sqrt[3]{10}$ .

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Step 5.1

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

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10}$ as $10^{\frac{1}{3}}$ .

$10 (10^{\frac{1}{3}} \sqrt[4]{10})$

Step 5.1.2

Rewrite $10^{\frac{1}{3}}$ as $10^{\frac{4}{12}}$ .

$10 (10^{\frac{4}{12}} \sqrt[4]{10})$

Step 5.1.3

Rewrite $10^{\frac{4}{12}}$ as $\sqrt[12]{10^{4}}$ .

$10 (\sqrt[12]{10^{4}} \sqrt[4]{10})$

Step 5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{10}$ as $10^{\frac{1}{4}}$ .

$10 (\sqrt[12]{10^{4}} \cdot 10^{\frac{1}{4}})$

Step 5.1.5

Rewrite $10^{\frac{1}{4}}$ as $10^{\frac{3}{12}}$ .

$10 (\sqrt[12]{10^{4}} \cdot 10^{\frac{3}{12}})$

Step 5.1.6

Rewrite $10^{\frac{3}{12}}$ as $\sqrt[12]{10^{3}}$ .

$10 (\sqrt[12]{10^{4}} \sqrt[12]{10^{3}})$

Step 5.2

Combine using the product rule for radicals.

$10 \sqrt[12]{10^{4} \cdot 10^{3}}$

Step 5.3

Multiply $10^{4}$ by $10^{3}$ by adding the exponents.

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Step 5.3.1

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

$10 \sqrt[12]{10^{4 + 3}}$

Step 5.3.2

Add $4$ and $3$ .

$10 \sqrt[12]{10^{7}}$

Step 6

Raise $10$ to the power of $7$ .

$10 \sqrt[12]{10000000}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt[12]{10000000}$

Decimal Form:

$38.31186849 \dots$