Pre-Algebra Examples

$\frac{3 \sqrt{10}}{5 \sqrt{25}}$

Step 1

Combine $\sqrt{10}$ and $\sqrt{25}$ into a single radical.

$\frac{3 \sqrt{\frac{10}{25}}}{5}$

Step 2

Cancel the common factor of $10$ and $25$ .

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Step 2.1

Factor $5$ out of $10$ .

$\frac{3 \sqrt{\frac{5 (2)}{25}}}{5}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $5$ out of $25$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

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

Step 3.3.6.4.2

Rewrite the expression.

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

Step 3.3.6.5

Evaluate the exponent.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

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

Step 3.4.2

Multiply $2$ by $5$ .

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

Step 4

Combine $3$ and $\frac{\sqrt{10}}{5}$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \sqrt{10}}{5} \cdot \frac{1}{5}$

Step 6

Multiply $\frac{3 \sqrt{10}}{5} \cdot \frac{1}{5}$ .

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Step 6.1

Multiply $\frac{3 \sqrt{10}}{5}$ by $\frac{1}{5}$ .

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

Step 6.2

Multiply $5$ by $5$ .

$\frac{3 \sqrt{10}}{25}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{25}$

Decimal Form:

$0.37947331 \dots$