Precalculus Examples

$\frac{4}{\sqrt{15}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{4}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{4 \sqrt{15}}{{\sqrt{15}}^{1} \sqrt{15}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{4 \sqrt{15}}{{\sqrt{15}}^{1} {\sqrt{15}}^{1}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.4

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

$\frac{4 \sqrt{15}}{{\sqrt{15}}^{1 + 1}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{4 \sqrt{15}}{{\sqrt{15}}^{2}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{4 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6.2

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

$\frac{4 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6.3

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

$\frac{4 \sqrt{15}}{15^{\frac{2}{2}}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{15}}{15^{\frac{2}{2}}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{15}}{15^{1}} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.2.6.5

Evaluate the exponent.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.3

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} - \frac{4}{\sqrt{3}}$

Step 1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} - \frac{4}{\sqrt{3}}$

Step 1.4.4

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}} - \frac{4}{\sqrt{3}}$

Step 1.4.5

Add $1$ and $1$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{2}} - \frac{4}{\sqrt{3}}$

Step 1.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} - \frac{4}{\sqrt{3}}$

Step 1.4.6.2

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} - \frac{4}{\sqrt{3}}$

Step 1.4.6.3

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}} - \frac{4}{\sqrt{3}}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}} - \frac{4}{\sqrt{3}}$

Step 1.4.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{5^{1}} - \frac{4}{\sqrt{3}}$

Step 1.4.6.5

Evaluate the exponent.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{3} \sqrt{5}}{5} - \frac{4}{\sqrt{3}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Combine using the product rule for radicals.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{5 \cdot 3}}{5} - \frac{4}{\sqrt{3}}$

Step 1.5.2

Multiply $5$ by $3$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4}{\sqrt{3}}$

Step 1.6

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - (\frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.7.2

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.7.3

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.7.4

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.7.5

Add $1$ and $1$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

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

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{3^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} - \frac{4 \sqrt{3}}{3}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{3 \sqrt{15}}{5}$ by $\frac{3}{3}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15}}{5} \cdot \frac{3}{3} - \frac{4 \sqrt{3}}{3}$

Step 2.2

Multiply $\frac{3 \sqrt{15}}{5}$ by $\frac{3}{3}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{5 \cdot 3} - \frac{4 \sqrt{3}}{3}$

Step 2.3

Multiply $\frac{4 \sqrt{3}}{3}$ by $\frac{5}{5}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{5 \cdot 3} - (\frac{4 \sqrt{3}}{3} \cdot \frac{5}{5})$

Step 2.4

Multiply $\frac{4 \sqrt{3}}{3}$ by $\frac{5}{5}$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{5 \cdot 3} - \frac{4 \sqrt{3} \cdot 5}{3 \cdot 5}$

Step 2.5

Reorder the factors of $5 \cdot 3$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{3 \cdot 5} - \frac{4 \sqrt{3} \cdot 5}{3 \cdot 5}$

Step 2.6

Multiply $3$ by $5$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{15} - \frac{4 \sqrt{3} \cdot 5}{3 \cdot 5}$

Step 2.7

Multiply $3$ by $5$ .

$\frac{4 \sqrt{15}}{15} + \frac{3 \sqrt{15} \cdot 3}{15} - \frac{4 \sqrt{3} \cdot 5}{15}$

Step 3

Combine the numerators over the common denominator.

$\frac{4 \sqrt{15} + 3 \sqrt{15} \cdot 3 - 4 \sqrt{3} \cdot 5}{15}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{4 \sqrt{15} + 9 \sqrt{15} - 4 \sqrt{3} \cdot 5}{15}$

Step 4.2

Multiply $5$ by $- 4$ .

$\frac{4 \sqrt{15} + 9 \sqrt{15} - 20 \sqrt{3}}{15}$

Step 5

Add $4 \sqrt{15}$ and $9 \sqrt{15}$ .

$\frac{13 \sqrt{15} - 20 \sqrt{3}}{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{13 \sqrt{15} - 20 \sqrt{3}}{15}$

Decimal Form:

$1.04718448 \dots$