Basic Math Examples

$\sqrt{\frac{1 + \frac{15}{21}}{2}}$

Step 1

Simplify the expression.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{21}{21} + \frac{15}{21}}{2}}$

Step 1.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{21 + 15}{21}}{2}}$

Step 1.3

Add $21$ and $15$ .

$\sqrt{\frac{\frac{36}{21}}{2}}$

Step 2

Cancel the common factor of $36$ and $21$ .

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Step 2.1

Factor $3$ out of $36$ .

$\sqrt{\frac{\frac{3 (12)}{21}}{2}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $21$ .

$\sqrt{\frac{\frac{3 \cdot 12}{3 \cdot 7}}{2}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{\frac{3 \cdot 12}{3 \cdot 7}}{2}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{\frac{12}{7}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{12}{7} \cdot \frac{1}{2}}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $12$ .

$\sqrt{\frac{2 (6)}{7} \cdot \frac{1}{2}}$

Step 4.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 6}{7} \cdot \frac{1}{2}}$

Step 4.3

Rewrite the expression.

$\sqrt{\frac{6}{7}}$

Step 5

Rewrite $\sqrt{\frac{6}{7}}$ as $\frac{\sqrt{6}}{\sqrt{7}}$ .

$\frac{\sqrt{6}}{\sqrt{7}}$

Step 6

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 7.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 7.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 7.4

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

$\frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 7.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{\sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{6} \sqrt{7}}{7^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{6} \sqrt{7}}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 7}}{7}$

Step 8.2

Multiply $6$ by $7$ .

$\frac{\sqrt{42}}{7}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{42}}{7}$

Decimal Form:

$0.92582009 \dots$