Algebra Examples

$\frac{9 - 3 \sqrt{15}}{12 - 4 \sqrt{15}}$

Step 1

Multiply $\frac{9 - 3 \sqrt{15}}{12 - 4 \sqrt{15}}$ by $\frac{12 + 4 \sqrt{15}}{12 + 4 \sqrt{15}}$ .

$\frac{9 - 3 \sqrt{15}}{12 - 4 \sqrt{15}} \cdot \frac{12 + 4 \sqrt{15}}{12 + 4 \sqrt{15}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{9 - 3 \sqrt{15}}{12 - 4 \sqrt{15}}$ by $\frac{12 + 4 \sqrt{15}}{12 + 4 \sqrt{15}}$ .

$\frac{(9 - 3 \sqrt{15}) (12 + 4 \sqrt{15})}{(12 - 4 \sqrt{15}) (12 + 4 \sqrt{15})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(9 - 3 \sqrt{15}) (12 + 4 \sqrt{15})}{144 + 48 \sqrt{15} - 48 \sqrt{15} - 16 {\sqrt{15}}^{2}}$

Step 2.3

Simplify.

$\frac{(9 - 3 \sqrt{15}) (12 + 4 \sqrt{15})}{- 96}$

Step 2.4

Cancel the common factor of $9 - 3 \sqrt{15}$ and $- 96$ .

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Step 2.4.1

Factor $3$ out of $(9 - 3 \sqrt{15}) (12 + 4 \sqrt{15})$ .

$\frac{3 ((3 - \sqrt{15}) (12 + 4 \sqrt{15}))}{- 96}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $- 96$ .

$\frac{3 ((3 - \sqrt{15}) (12 + 4 \sqrt{15}))}{3 (- 32)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{3 ((3 - \sqrt{15}) (12 + 4 \sqrt{15}))}{3 \cdot - 32}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(3 - \sqrt{15}) (12 + 4 \sqrt{15})}{- 32}$

Step 2.5

Cancel the common factor of $12 + 4 \sqrt{15}$ and $- 32$ .

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Step 2.5.1

Factor $4$ out of $(3 - \sqrt{15}) (12 + 4 \sqrt{15})$ .

$\frac{4 ((3 - \sqrt{15}) (3 + \sqrt{15}))}{- 32}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $4$ out of $- 32$ .

$\frac{4 ((3 - \sqrt{15}) (3 + \sqrt{15}))}{4 (- 8)}$

Step 2.5.2.2

Cancel the common factor.

$\frac{4 ((3 - \sqrt{15}) (3 + \sqrt{15}))}{4 \cdot - 8}$

Step 2.5.2.3

Rewrite the expression.

$\frac{(3 - \sqrt{15}) (3 + \sqrt{15})}{- 8}$

Step 3

Simplify the numerator.

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Step 3.1

Expand $(3 - \sqrt{15}) (3 + \sqrt{15})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{3 (3 + \sqrt{15}) - \sqrt{15} (3 + \sqrt{15})}{- 8}$

Step 3.1.2

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{15} - \sqrt{15} (3 + \sqrt{15})}{- 8}$

Step 3.1.3

Apply the distributive property.

$\frac{3 \cdot 3 + 3 \sqrt{15} - \sqrt{15} \cdot 3 - \sqrt{15} \sqrt{15}}{- 8}$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 \sqrt{15} - \sqrt{15} \cdot 3 - \sqrt{15} \sqrt{15}}{- 8}$

Step 3.2.1.2

Multiply $3$ by $- 1$ .

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - \sqrt{15} \sqrt{15}}{- 8}$

Step 3.2.1.3

Multiply $- \sqrt{15} \sqrt{15}$ .

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Step 3.2.1.3.1

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - ({\sqrt{15}}^{1} \sqrt{15})}{- 8}$

Step 3.2.1.3.2

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{- 8}$

Step 3.2.1.3.3

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - {\sqrt{15}}^{1 + 1}}{- 8}$

Step 3.2.1.3.4

Add $1$ and $1$ .

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - {\sqrt{15}}^{2}}{- 8}$

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}}{- 8}$

Step 3.2.1.4.2

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}}{- 8}$

Step 3.2.1.4.3

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

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 15^{\frac{2}{2}}}{- 8}$

Step 3.2.1.4.4

Cancel the common factor of $2$ .

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Step 3.2.1.4.4.1

Cancel the common factor.

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 15^{\frac{2}{2}}}{- 8}$

Step 3.2.1.4.4.2

Rewrite the expression.

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 15^{1}}{- 8}$

Step 3.2.1.4.5

Evaluate the exponent.

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 1 \cdot 15}{- 8}$

Step 3.2.1.5

Multiply $- 1$ by $15$ .

$\frac{9 + 3 \sqrt{15} - 3 \sqrt{15} - 15}{- 8}$

Step 3.2.2

Subtract $15$ from $9$ .

$\frac{- 6 + 3 \sqrt{15} - 3 \sqrt{15}}{- 8}$

Step 3.2.3

Subtract $3 \sqrt{15}$ from $3 \sqrt{15}$ .

$\frac{- 6 + 0}{- 8}$

Step 3.2.4

Add $- 6$ and $0$ .

$\frac{- 6}{- 8}$

Step 4

Cancel the common factor of $- 6$ and $- 8$ .

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Step 4.1

Factor $- 2$ out of $- 6$ .

$\frac{- 2 (3)}{- 8}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $- 2$ out of $- 8$ .

$\frac{- 2 \cdot 3}{- 2 \cdot 4}$

Step 4.2.2

Cancel the common factor.

$\frac{- 2 \cdot 3}{- 2 \cdot 4}$

Step 4.2.3

Rewrite the expression.

$\frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$