Basic Math Examples

$\sqrt{\frac{4}{7}} \cdot \sqrt{\frac{2}{35}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{\frac{4}{7} \cdot \frac{2}{35}}$

Step 2

Multiply $\frac{4}{7}$ by $\frac{2}{35}$ .

$\sqrt{\frac{4 \cdot 2}{7 \cdot 35}}$

Step 3

Multiply.

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Step 3.1

Multiply $4$ by $2$ .

$\sqrt{\frac{8}{7 \cdot 35}}$

Step 3.2

Multiply $7$ by $35$ .

$\sqrt{\frac{8}{245}}$

Step 4

Rewrite $\sqrt{\frac{8}{245}}$ as $\frac{\sqrt{8}}{\sqrt{245}}$ .

$\frac{\sqrt{8}}{\sqrt{245}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt{245}}$

Step 5.1.2

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

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{245}}$

Step 5.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{\sqrt{245}}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $245$ as $7^{2} \cdot 5$ .

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Step 6.1.1

Factor $49$ out of $245$ .

$\frac{2 \sqrt{2}}{\sqrt{49 (5)}}$

Step 6.1.2

Rewrite $49$ as $7^{2}$ .

$\frac{2 \sqrt{2}}{\sqrt{7^{2} \cdot 5}}$

Step 6.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{7 \sqrt{5}}$

Step 7

Multiply $\frac{2 \sqrt{2}}{7 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{2}}{7 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{2 \sqrt{2}}{7 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{2} \sqrt{5}}{7 \sqrt{5} \sqrt{5}}$

Step 8.2

Move $\sqrt{5}$ .

$\frac{2 \sqrt{2} \sqrt{5}}{7 (\sqrt{5} \sqrt{5})}$

Step 8.3

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 8.4

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 8.5

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 {\sqrt{5}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{2 \sqrt{2} \sqrt{5}}{7 {\sqrt{5}}^{2}}$

Step 8.7

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

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Step 8.7.1

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 {(5^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$\frac{2 \sqrt{2} \sqrt{5}}{7 \cdot 5^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{2} \sqrt{5}}{7 \cdot 5^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{2 \sqrt{2} \sqrt{5}}{7 \cdot 5^{1}}$

Step 8.7.5

Evaluate the exponent.

$\frac{2 \sqrt{2} \sqrt{5}}{7 \cdot 5}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{5 \cdot 2}}{7 \cdot 5}$

Step 9.2

Multiply $5$ by $2$ .

$\frac{2 \sqrt{10}}{7 \cdot 5}$

Step 10

Multiply $7$ by $5$ .

$\frac{2 \sqrt{10}}{35}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{10}}{35}$

Decimal Form:

$0.18070158 \dots$