Algebra Examples

$\sqrt[5]{10^{2}} \cdot \sqrt[6]{10^{5}}$

Step 1

Raise $10$ to the power of $2$ .

$\sqrt[5]{100} \cdot \sqrt[6]{10^{5}}$

Step 2

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

$\sqrt[5]{100} \cdot \sqrt[6]{100000}$

Step 3

Multiply $\sqrt[5]{100} \sqrt[6]{100000}$ .

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Step 3.1

Rewrite the expression using the least common index of $30$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{100}$ as $100^{\frac{1}{5}}$ .

$100^{\frac{1}{5}} \sqrt[6]{100000}$

Step 3.1.2

Rewrite $100^{\frac{1}{5}}$ as $100^{\frac{6}{30}}$ .

$100^{\frac{6}{30}} \sqrt[6]{100000}$

Step 3.1.3

Rewrite $100^{\frac{6}{30}}$ as $\sqrt[30]{100^{6}}$ .

$\sqrt[30]{100^{6}} \sqrt[6]{100000}$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{100000}$ as $100000^{\frac{1}{6}}$ .

$\sqrt[30]{100^{6}} \cdot 100000^{\frac{1}{6}}$

Step 3.1.5

Rewrite $100000^{\frac{1}{6}}$ as $100000^{\frac{5}{30}}$ .

$\sqrt[30]{100^{6}} \cdot 100000^{\frac{5}{30}}$

Step 3.1.6

Rewrite $100000^{\frac{5}{30}}$ as $\sqrt[30]{100000^{5}}$ .

$\sqrt[30]{100^{6}} \sqrt[30]{100000^{5}}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[30]{100^{6} \cdot 100000^{5}}$

Step 3.3

Rewrite $100$ as $10^{2}$ .

$\sqrt[30]{{(10^{2})}^{6} \cdot 100000^{5}}$

Step 3.4

Multiply the exponents in ${(10^{2})}^{6}$ .

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Step 3.4.1

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

$\sqrt[30]{10^{2 \cdot 6} \cdot 100000^{5}}$

Step 3.4.2

Multiply $2$ by $6$ .

$\sqrt[30]{10^{12} \cdot 100000^{5}}$

Step 3.5

Rewrite $100000$ as $10^{5}$ .

$\sqrt[30]{10^{12} \cdot {(10^{5})}^{5}}$

Step 3.6

Multiply the exponents in ${(10^{5})}^{5}$ .

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Step 3.6.1

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

$\sqrt[30]{10^{12} \cdot 10^{5 \cdot 5}}$

Step 3.6.2

Multiply $5$ by $5$ .

$\sqrt[30]{10^{12} \cdot 10^{25}}$

Step 3.7

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

$\sqrt[30]{10^{12 + 25}}$

Step 3.8

Add $12$ and $25$ .

$\sqrt[30]{10^{37}}$

Step 4

Factor out $10^{30}$ .

$\sqrt[30]{10^{30} \cdot 10^{7}}$

Step 5

Pull terms out from under the radical.

$| 10 | \sqrt[30]{10^{7}}$

Step 6

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

$10 \sqrt[30]{10^{7}}$

Step 7

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

$10 \sqrt[30]{10000000}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt[30]{10000000}$

Decimal Form:

$17.11328304 \dots$