Algebra Examples

$(\sqrt{5 a}) \cdot (\sqrt{2 a}) \div (3 \sqrt{10 a^{2}})$

Step 1

Rewrite the division as a fraction.

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

Step 2

Combine $\sqrt{5 a}$ and $\sqrt{10 a^{2}}$ into a single radical.

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $5$ and $10$ .

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Step 3.1.1

Factor $5$ out of $5 a$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $5$ out of $10 a^{2}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $a$ and $a^{2}$ .

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Step 3.2.1

Raise $a$ to the power of $1$ .

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

Step 3.2.2

Factor $a$ out of $a^{1}$ .

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

Step 3.2.3

Cancel the common factors.

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Step 3.2.3.1

Factor $a$ out of $2 a^{2}$ .

$\frac{\sqrt{\frac{a \cdot 1}{a (2 a)}} \cdot \sqrt{2 a}}{3}$

Step 3.2.3.2

Cancel the common factor.

$\frac{\sqrt{\frac{a \cdot 1}{a (2 a)}} \cdot \sqrt{2 a}}{3}$

Step 3.2.3.3

Rewrite the expression.

$\frac{\sqrt{\frac{1}{2 a}} \cdot \sqrt{2 a}}{3}$

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{\frac{1}{2 a} (2 a)}}{3}$

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the numerator.

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Step 5.1

Reduce the expression $\frac{2 a}{2 a}$ by cancelling the common factors.

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

$\frac{\sqrt{\frac{a}{a}}}{3}$

Step 5.2

Cancel the common factor of $a$ .

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Step 5.2.1

Cancel the common factor.

$\frac{\sqrt{\frac{a}{a}}}{3}$

Step 5.2.2

Rewrite the expression.

$\frac{\sqrt{1}}{3}$

Step 5.3

Any root of $1$ is $1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{\frac{a}{a}}{3}$

Step 5.3.2

Rewrite the expression.

$\frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$