Algebra Examples

$\frac{15 \sqrt{3} - 5 \sqrt{18}}{3 \sqrt{12}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 1.1.1

Factor $9$ out of $18$ .

$\frac{15 \sqrt{3} - 5 \sqrt{9 (2)}}{3 \sqrt{12}}$

Step 1.1.2

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

$\frac{15 \sqrt{3} - 5 \sqrt{3^{2} \cdot 2}}{3 \sqrt{12}}$

Step 1.2

Pull terms out from under the radical.

$\frac{15 \sqrt{3} - 5 (3 \sqrt{2})}{3 \sqrt{12}}$

Step 1.3

Multiply $3$ by $- 5$ .

$\frac{15 \sqrt{3} - 15 \sqrt{2}}{3 \sqrt{12}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $12$ .

$\frac{15 \sqrt{3} - 15 \sqrt{2}}{3 \sqrt{4 (3)}}$

Step 2.1.2

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

$\frac{15 \sqrt{3} - 15 \sqrt{2}}{3 \sqrt{2^{2} \cdot 3}}$

Step 2.2

Pull terms out from under the radical.

$\frac{15 \sqrt{3} - 15 \sqrt{2}}{3 \cdot 2 \sqrt{3}}$

Step 2.3

Multiply $3$ by $2$ .

$\frac{15 \sqrt{3} - 15 \sqrt{2}}{6 \sqrt{3}}$

Step 3

Cancel the common factor of $15 \sqrt{3} - 15 \sqrt{2}$ and $6$ .

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Step 3.1

Factor $3$ out of $15 \sqrt{3}$ .

$\frac{3 (5 \sqrt{3}) - 15 \sqrt{2}}{6 \sqrt{3}}$

Step 3.2

Factor $3$ out of $- 15 \sqrt{2}$ .

$\frac{3 (5 \sqrt{3}) + 3 (- 5 \sqrt{2})}{6 \sqrt{3}}$

Step 3.3

Factor $3$ out of $3 (5 \sqrt{3}) + 3 (- 5 \sqrt{2})$ .

$\frac{3 (5 \sqrt{3} - 5 \sqrt{2})}{6 \sqrt{3}}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $3$ out of $6 \sqrt{3}$ .

$\frac{3 (5 \sqrt{3} - 5 \sqrt{2})}{3 (2 \sqrt{3})}$

Step 3.4.2

Cancel the common factor.

$\frac{3 (5 \sqrt{3} - 5 \sqrt{2})}{3 (2 \sqrt{3})}$

Step 3.4.3

Rewrite the expression.

$\frac{5 \sqrt{3} - 5 \sqrt{2}}{2 \sqrt{3}}$

Step 4

Multiply $\frac{5 \sqrt{3} - 5 \sqrt{2}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{5 \sqrt{3} - 5 \sqrt{2}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{5 \sqrt{3} - 5 \sqrt{2}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 5.2

Move $\sqrt{3}$ .

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 5.3

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

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 5.4

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

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 5.5

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

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

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

Step 5.7

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

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Step 5.7.1

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

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

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

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{2 \cdot 3^{1}}$

Step 5.7.5

Evaluate the exponent.

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

Step 6

Multiply $2$ by $3$ .

$\frac{(5 \sqrt{3} - 5 \sqrt{2}) \sqrt{3}}{6}$

Step 7

Apply the distributive property.

$\frac{5 \sqrt{3} \sqrt{3} - 5 \sqrt{2} \sqrt{3}}{6}$

Step 8

Multiply $5 \sqrt{3} \sqrt{3}$ .

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Step 8.1

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

$\frac{5 ({\sqrt{3}}^{1} \sqrt{3}) - 5 \sqrt{2} \sqrt{3}}{6}$

Step 8.2

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

$\frac{5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 5 \sqrt{2} \sqrt{3}}{6}$

Step 8.3

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

$\frac{5 {\sqrt{3}}^{1 + 1} - 5 \sqrt{2} \sqrt{3}}{6}$

Step 8.4

Add $1$ and $1$ .

$\frac{5 {\sqrt{3}}^{2} - 5 \sqrt{2} \sqrt{3}}{6}$

Step 9

Multiply $- 5 \sqrt{2} \sqrt{3}$ .

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Step 9.1

Combine using the product rule for radicals.

$\frac{5 {\sqrt{3}}^{2} - 5 \sqrt{3 \cdot 2}}{6}$

Step 9.2

Multiply $3$ by $2$ .

$\frac{5 {\sqrt{3}}^{2} - 5 \sqrt{6}}{6}$

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

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

$\frac{5 {(3^{\frac{1}{2}})}^{2} - 5 \sqrt{6}}{6}$

Step 10.1.2

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

$\frac{5 \cdot 3^{\frac{1}{2} \cdot 2} - 5 \sqrt{6}}{6}$

Step 10.1.3

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

$\frac{5 \cdot 3^{\frac{2}{2}} - 5 \sqrt{6}}{6}$

Step 10.1.4

Cancel the common factor of $2$ .

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Step 10.1.4.1

Cancel the common factor.

$\frac{5 \cdot 3^{\frac{2}{2}} - 5 \sqrt{6}}{6}$

Step 10.1.4.2

Rewrite the expression.

$\frac{5 \cdot 3^{1} - 5 \sqrt{6}}{6}$

Step 10.1.5

Evaluate the exponent.

$\frac{5 \cdot 3 - 5 \sqrt{6}}{6}$

Step 10.2

Multiply $5$ by $3$ .

$\frac{15 - 5 \sqrt{6}}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{15 - 5 \sqrt{6}}{6}$

Decimal Form:

$0.45875854 \dots$