Basic Math Examples

$\sqrt{108 + \sqrt{147}} \div \sqrt{3}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{108 + \sqrt{147}}}{\sqrt{3}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 2.1.1

Factor $49$ out of $147$ .

$\frac{\sqrt{108 + \sqrt{49 (3)}}}{\sqrt{3}}$

Step 2.1.2

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

$\frac{\sqrt{108 + \sqrt{7^{2} \cdot 3}}}{\sqrt{3}}$

Step 2.2

Pull terms out from under the radical.

$\frac{\sqrt{108 + 7 \sqrt{3}}}{\sqrt{3}}$

Step 3

Multiply $\frac{\sqrt{108 + 7 \sqrt{3}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{108 + 7 \sqrt{3}}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{108 + 7 \sqrt{3}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.4

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

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{3^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{\sqrt{108 + 7 \sqrt{3}} \sqrt{3}}{3}$

Step 5

Combine using the product rule for radicals.

$\frac{\sqrt{(108 + 7 \sqrt{3}) \cdot 3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(108 + 7 \sqrt{3}) \cdot 3}}{3}$

Decimal Form:

$6.32783153 \dots$