Finite Math Examples

$\sqrt{(10) (\frac{2}{5}) (\frac{3}{5})}$

Step 1

Simplify terms.

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Step 1.1

Combine $10$ and $\frac{2}{5}$ .

$\sqrt{\frac{10 \cdot 2}{5} \cdot \frac{3}{5}}$

Step 1.2

Multiply $\frac{10 \cdot 2}{5}$ by $\frac{3}{5}$ .

$\sqrt{\frac{10 \cdot 2 \cdot 3}{5 \cdot 5}}$

Step 1.3

Reduce the expression $\frac{10 \cdot 2 \cdot 3}{5 \cdot 5}$ by cancelling the common factors.

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Step 1.3.1

Factor $5$ out of $10 \cdot 2 \cdot 3$ .

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

Step 1.3.2

Factor $5$ out of $5 \cdot 5$ .

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

Step 1.3.3

Cancel the common factor.

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

Step 1.3.4

Rewrite the expression.

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

Step 2

Simplify the numerator.

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Step 2.1

Multiply $2$ by $2$ .

$\sqrt{\frac{4 \cdot 3}{5}}$

Step 2.2

Multiply $4$ by $3$ .

$\sqrt{\frac{12}{5}}$

Step 3

Rewrite $\sqrt{\frac{12}{5}}$ as $\frac{\sqrt{12}}{\sqrt{5}}$ .

$\frac{\sqrt{12}}{\sqrt{5}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)}}{\sqrt{5}}$

Step 4.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $5$ by $3$ .

$\frac{2 \sqrt{15}}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{15}}{5}$

Decimal Form:

$1.54919333 \dots$