Linear Algebra Examples

$10 \sqrt{\frac{21000}{10000}}$

Step 1

Cancel the common factor of $21000$ and $10000$ .

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Step 1.1

Factor $1000$ out of $21000$ .

$10 \sqrt{\frac{1000 (21)}{10000}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $1000$ out of $10000$ .

$10 \sqrt{\frac{1000 \cdot 21}{1000 \cdot 10}}$

Step 1.2.2

Cancel the common factor.

$10 \sqrt{\frac{1000 \cdot 21}{1000 \cdot 10}}$

Step 1.2.3

Rewrite the expression.

$10 \sqrt{\frac{21}{10}}$

Step 2

Rewrite $\sqrt{\frac{21}{10}}$ as $\frac{\sqrt{21}}{\sqrt{10}}$ .

$10 \frac{\sqrt{21}}{\sqrt{10}}$

Step 3

Multiply $\frac{\sqrt{21}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$10 (\frac{\sqrt{21}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{21}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$10 \frac{\sqrt{21} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.2

Raise $\sqrt{10}$ to the power of $1$ .

$10 \frac{\sqrt{21} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 4.3

Raise $\sqrt{10}$ to the power of $1$ .

$10 \frac{\sqrt{21} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 4.4

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

$10 \frac{\sqrt{21} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$10 \frac{\sqrt{21} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$10 \frac{\sqrt{21} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$10 \frac{\sqrt{21} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$10 \frac{\sqrt{21} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$10 \frac{\sqrt{21} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$10 \frac{\sqrt{21} \sqrt{10}}{10^{1}}$

Step 4.6.5

Evaluate the exponent.

$10 \frac{\sqrt{21} \sqrt{10}}{10}$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$10 \frac{\sqrt{21 \cdot 10}}{10}$

Step 5.2

Multiply $21$ by $10$ .

$10 \frac{\sqrt{210}}{10}$

Step 6

Cancel the common factor of $10$ .

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Step 6.1

Cancel the common factor.

$10 \frac{\sqrt{210}}{10}$

Step 6.2

Rewrite the expression.

$\sqrt{210}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{210}$

Decimal Form:

$14.49137674 \dots$