Basic Math Examples

$M = \frac{30}{\sqrt{12} + \sqrt{27} + \sqrt{75}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

$M = \frac{30}{\sqrt{4 (3)} + \sqrt{27} + \sqrt{75}}$

Step 1.1.2

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

$M = \frac{30}{\sqrt{2^{2} \cdot 3} + \sqrt{27} + \sqrt{75}}$

Step 1.2

Pull terms out from under the radical.

$M = \frac{30}{2 \sqrt{3} + \sqrt{27} + \sqrt{75}}$

Step 1.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.3.1

Factor $9$ out of $27$ .

$M = \frac{30}{2 \sqrt{3} + \sqrt{9 (3)} + \sqrt{75}}$

Step 1.3.2

Rewrite $9$ as $3^{2}$ .

$M = \frac{30}{2 \sqrt{3} + \sqrt{3^{2} \cdot 3} + \sqrt{75}}$

Step 1.4

Pull terms out from under the radical.

$M = \frac{30}{2 \sqrt{3} + 3 \sqrt{3} + \sqrt{75}}$

Step 1.5

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.5.1

Factor $25$ out of $75$ .

$M = \frac{30}{2 \sqrt{3} + 3 \sqrt{3} + \sqrt{25 (3)}}$

Step 1.5.2

Rewrite $25$ as $5^{2}$ .

$M = \frac{30}{2 \sqrt{3} + 3 \sqrt{3} + \sqrt{5^{2} \cdot 3}}$

Step 1.6

Pull terms out from under the radical.

$M = \frac{30}{2 \sqrt{3} + 3 \sqrt{3} + 5 \sqrt{3}}$

Step 1.7

Add $2 \sqrt{3}$ and $3 \sqrt{3}$ .

$M = \frac{30}{5 \sqrt{3} + 5 \sqrt{3}}$

Step 1.8

Add $5 \sqrt{3}$ and $5 \sqrt{3}$ .

$M = \frac{30}{10 \sqrt{3}}$

Step 2

Cancel the common factor of $30$ and $10$ .

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Step 2.1

Factor $10$ out of $30$ .

$M = \frac{10 \cdot 3}{10 \sqrt{3}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $10$ out of $10 \sqrt{3}$ .

$M = \frac{10 \cdot 3}{10 (\sqrt{3})}$

Step 2.2.2

Cancel the common factor.

$M = \frac{10 \cdot 3}{10 \sqrt{3}}$

Step 2.2.3

Rewrite the expression.

$M = \frac{3}{\sqrt{3}}$

Step 3

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$M = \frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$M = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2

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

$M = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.3

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

$M = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.4

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

$M = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$M = \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}}$

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 $3^{\frac{1}{2}}$ .

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

Step 4.6.2

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

$M = \frac{3 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$M = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$M = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$M = \frac{3 \sqrt{3}}{3^{1}}$

Step 4.6.5

Evaluate the exponent.

$M = \frac{3 \sqrt{3}}{3}$

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Cancel the common factor.

$M = \frac{3 \sqrt{3}}{3}$

Step 5.2

Divide $\sqrt{3}$ by $1$ .

$M = \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$M = \sqrt{3}$

Decimal Form:

$M = 1.73205080 \dots$