Algebra Examples

$\frac{3 - 3 \sqrt{3 a}}{4 \sqrt{8 a}}$

Step 1

Factor $3$ out of $3 - 3 \sqrt{3 a}$ .

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Step 1.1

Factor $3$ out of $3$ .

$\frac{3 (1) - 3 \sqrt{3 a}}{4 \sqrt{8 a}}$

Step 1.2

Factor $3$ out of $- 3 \sqrt{3 a}$ .

$\frac{3 (1) + 3 (- \sqrt{3 a})}{4 \sqrt{8 a}}$

Step 1.3

Factor $3$ out of $3 (1) + 3 (- \sqrt{3 a})$ .

$\frac{3 (1 - \sqrt{3 a})}{4 \sqrt{8 a}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $8 a$ as $2^{2} \cdot (2 a)$ .

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Step 2.1.1

Factor $4$ out of $8$ .

$\frac{3 (1 - \sqrt{3 a})}{4 \sqrt{4 (2) a}}$

Step 2.1.2

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

$\frac{3 (1 - \sqrt{3 a})}{4 \sqrt{2^{2} \cdot 2 a}}$

Step 2.1.3

Add parentheses.

$\frac{3 (1 - \sqrt{3 a})}{4 \sqrt{2^{2} \cdot (2 a)}}$

Step 2.2

Pull terms out from under the radical.

$\frac{3 (1 - \sqrt{3 a})}{4 \cdot 2 \sqrt{2 a}}$

Step 2.3

Multiply $4$ by $2$ .

$\frac{3 (1 - \sqrt{3 a})}{8 \sqrt{2 a}}$

Step 3

Multiply $\frac{3 (1 - \sqrt{3 a})}{8 \sqrt{2 a}}$ by $\frac{\sqrt{2 a}}{\sqrt{2 a}}$ .

$\frac{3 (1 - \sqrt{3 a})}{8 \sqrt{2 a}} \cdot \frac{\sqrt{2 a}}{\sqrt{2 a}}$

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{3 (1 - \sqrt{3 a})}{8 \sqrt{2 a}}$ by $\frac{\sqrt{2 a}}{\sqrt{2 a}}$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 \sqrt{2 a} \sqrt{2 a}}$

Step 4.1.2

Move $\sqrt{2 a}$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 (\sqrt{2 a} \sqrt{2 a})}$

Step 4.1.3

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 ({\sqrt{2 a}}^{1} \sqrt{2 a})}$

Step 4.1.4

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 ({\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1})}$

Step 4.1.5

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

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

Step 4.1.6

Add $1$ and $1$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {\sqrt{2 a}}^{2}}$

Step 4.1.7

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

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Step 4.1.7.1

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

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {({(2 a)}^{\frac{1}{2}})}^{2}}$

Step 4.1.7.2

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

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {(2 a)}^{\frac{1}{2} \cdot 2}}$

Step 4.1.7.3

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

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {(2 a)}^{\frac{2}{2}}}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {(2 a)}^{\frac{2}{2}}}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 {(2 a)}^{1}}$

Step 4.1.7.5

Simplify.

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{8 (2 a)}$

Step 4.2

Simplify by multiplying through.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Multiply $8$ by $2$ .

$\frac{3 (1 - \sqrt{3 a}) \sqrt{2 a}}{16 a}$

Step 4.2.1.2

Group $\sqrt{2 a}$ and $1 - \sqrt{3 a}$ together.

$\frac{3 (\sqrt{2 a} (1 - \sqrt{3 a}))}{16 a}$

Step 4.2.2

Apply the distributive property.

$\frac{3 (\sqrt{2 a} \cdot 1 + \sqrt{2 a} (- \sqrt{3 a}))}{16 a}$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $\sqrt{2 a}$ by $1$ .

$\frac{3 (\sqrt{2 a} + \sqrt{2 a} (- \sqrt{3 a}))}{16 a}$

Step 4.2.3.2

Rewrite using the commutative property of multiplication.

$\frac{3 (\sqrt{2 a} - \sqrt{2 a} \sqrt{3 a})}{16 a}$

Step 4.3

Simplify each term.

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Step 4.3.1

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

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Step 4.3.1.1

Combine using the product rule for radicals.

$\frac{3 (\sqrt{2 a} - \sqrt{3 a (2 a)})}{16 a}$

Step 4.3.1.2

Multiply $2$ by $3$ .

$\frac{3 (\sqrt{2 a} - \sqrt{6 a \cdot a})}{16 a}$

Step 4.3.1.3

Raise $a$ to the power of $1$ .

$\frac{3 (\sqrt{2 a} - \sqrt{6 (a^{1} a)})}{16 a}$

Step 4.3.1.4

Raise $a$ to the power of $1$ .

$\frac{3 (\sqrt{2 a} - \sqrt{6 (a^{1} a^{1})})}{16 a}$

Step 4.3.1.5

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

$\frac{3 (\sqrt{2 a} - \sqrt{6 a^{1 + 1}})}{16 a}$

Step 4.3.1.6

Add $1$ and $1$ .

$\frac{3 (\sqrt{2 a} - \sqrt{6 a^{2}})}{16 a}$

Step 4.3.2

Reorder $6$ and $a^{2}$ .

$\frac{3 (\sqrt{2 a} - \sqrt{a^{2} \cdot 6})}{16 a}$

Step 4.3.3

Pull terms out from under the radical.

$\frac{3 (\sqrt{2 a} - a \sqrt{6})}{16 a}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{3 ({(2 a)}^{\frac{1}{2}} - a \sqrt{6})}{16 a}$

Step 5.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{3 ({(2 a)}^{\frac{1}{2}} - a \cdot 6^{\frac{1}{2}})}{16 a}$

Step 5.3

Apply the product rule to $2 a$ .

$\frac{3 (2^{\frac{1}{2}} a^{\frac{1}{2}} - a \cdot 6^{\frac{1}{2}})}{16 a}$

Step 5.4

Rewrite $2^{\frac{1}{2}} a^{\frac{1}{2}} - a \cdot 6^{\frac{1}{2}}$ in a factored form.

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Step 5.4.1

Factor $a^{\frac{1}{2}}$ out of $2^{\frac{1}{2}} a^{\frac{1}{2}} - a \cdot 6^{\frac{1}{2}}$ .

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Step 5.4.1.1

Move $a$ .

$\frac{3 (2^{\frac{1}{2}} a^{\frac{1}{2}} - 1 \cdot 6^{\frac{1}{2}} a)}{16 a}$

Step 5.4.1.2

Factor $a^{\frac{1}{2}}$ out of $2^{\frac{1}{2}} a^{\frac{1}{2}}$ .

$\frac{3 (a^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} - 1 \cdot 6^{\frac{1}{2}} a)}{16 a}$

Step 5.4.1.3

Factor $a^{\frac{1}{2}}$ out of $- 1 \cdot 6^{\frac{1}{2}} a$ .

$\frac{3 (a^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} + a^{\frac{1}{2}} (- 1 \cdot 6^{\frac{1}{2}} a^{\frac{1}{2}}))}{16 a}$

Step 5.4.1.4

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} + a^{\frac{1}{2}} (- 1 \cdot 6^{\frac{1}{2}} a^{\frac{1}{2}})$ .

$\frac{3 (a^{\frac{1}{2}} (2^{\frac{1}{2}} - 1 \cdot 6^{\frac{1}{2}} a^{\frac{1}{2}}))}{16 a}$

Step 5.4.2

Rewrite $- 1 \cdot 6^{\frac{1}{2}}$ as $- 6^{\frac{1}{2}}$ .

$\frac{3 (a^{\frac{1}{2}} (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}}))}{16 a}$

$\frac{3 a^{\frac{1}{2}} (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a}$

Step 6

Move $a^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a \cdot a^{- \frac{1}{2}}}$

Step 7

Multiply $a$ by $a^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.1

Move $a^{- \frac{1}{2}}$ .

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 (a^{- \frac{1}{2}} a)}$

Step 7.2

Multiply $a^{- \frac{1}{2}}$ by $a$ .

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Step 7.2.1

Raise $a$ to the power of $1$ .

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 (a^{- \frac{1}{2}} a^{1})}$

Step 7.2.2

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

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a^{- \frac{1}{2} + 1}}$

Step 7.3

Write $1$ as a fraction with a common denominator.

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a^{- \frac{1}{2} + \frac{2}{2}}}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a^{\frac{- 1 + 2}{2}}}$

Step 7.5

Add $- 1$ and $2$ .

$\frac{3 (2^{\frac{1}{2}} - 6^{\frac{1}{2}} a^{\frac{1}{2}})}{16 a^{\frac{1}{2}}}$