Basic Math Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

Multiply $3$ by $5$ .

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

Step 5

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

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Step 5.1

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

$\frac{3 \sqrt{15} - 2 ({\sqrt{3}}^{1} \sqrt{3})}{3}$

Step 5.2

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

$\frac{3 \sqrt{15} - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{3}$

Step 5.3

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

$\frac{3 \sqrt{15} - 2 {\sqrt{3}}^{1 + 1}}{3}$

Step 5.4

Add $1$ and $1$ .

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

Step 6

Simplify each term.

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Step 6.1

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

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Step 6.1.1

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

$\frac{3 \sqrt{15} - 2 {(3^{\frac{1}{2}})}^{2}}{3}$

Step 6.1.2

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

$\frac{3 \sqrt{15} - 2 \cdot 3^{\frac{1}{2} \cdot 2}}{3}$

Step 6.1.3

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

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

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Cancel the common factor.

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

Step 6.1.4.2

Rewrite the expression.

$\frac{3 \sqrt{15} - 2 \cdot 3^{1}}{3}$

Step 6.1.5

Evaluate the exponent.

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

Step 6.2

Multiply $- 2$ by $3$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Factor $3$ out of $- 6$ .

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

Step 7.3

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

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

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $3$ out of $3$ .

$\frac{3 (\sqrt{15} - 2)}{3 (1)}$

Step 7.4.2

Cancel the common factor.

$\frac{3 (\sqrt{15} - 2)}{3 \cdot 1}$

Step 7.4.3

Rewrite the expression.

$\frac{\sqrt{15} - 2}{1}$

Step 7.4.4

Divide $\sqrt{15} - 2$ by $1$ .

$\sqrt{15} - 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15} - 2$

Decimal Form:

$1.87298334 \dots$