Finite Math Examples

$\sqrt{8} \cdot \sqrt{\frac{9}{20}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{8 \cdot \frac{9}{20}}$

Step 2

Combine $8$ and $\frac{9}{20}$ .

$\sqrt{\frac{8 \cdot 9}{20}}$

Step 3

Reduce the expression $\frac{8 \cdot 9}{20}$ by cancelling the common factors.

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Step 3.1

Factor $4$ out of $8 \cdot 9$ .

$\sqrt{\frac{4 (2 \cdot 9)}{20}}$

Step 3.2

Factor $4$ out of $20$ .

$\sqrt{\frac{4 (2 \cdot 9)}{4 (5)}}$

Step 3.3

Cancel the common factor.

$\sqrt{\frac{4 (2 \cdot 9)}{4 \cdot 5}}$

Step 3.4

Rewrite the expression.

$\sqrt{\frac{2 \cdot 9}{5}}$

Step 4

Multiply $2$ by $9$ .

$\sqrt{\frac{18}{5}}$

Step 5

Rewrite $\sqrt{\frac{18}{5}}$ as $\frac{\sqrt{18}}{\sqrt{5}}$ .

$\frac{\sqrt{18}}{\sqrt{5}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)}}{\sqrt{5}}$

Step 6.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2}}{\sqrt{5}}$

Step 6.2

Pull terms out from under the radical.

$\frac{3 \sqrt{2}}{\sqrt{5}}$

Step 7

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 8.4

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

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{2} \sqrt{5}}{5^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{3 \sqrt{2} \sqrt{5}}{5}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{5 \cdot 2}}{5}$

Step 9.2

Multiply $5$ by $2$ .

$\frac{3 \sqrt{10}}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{5}$

Decimal Form:

$1.89736659 \dots$