Algebra Examples

$\frac{16 \sqrt{a^{5}} - 8 \sqrt{a^{3}} + 2 \sqrt{a}}{2 \sqrt{a}}$

Step 1

Cancel the common factor of $16 \sqrt{a^{5}} - 8 \sqrt{a^{3}} + 2 \sqrt{a}$ and $2$ .

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Step 1.1

Factor $2$ out of $16 \sqrt{a^{5}}$ .

$\frac{2 (8 \sqrt{a^{5}}) - 8 \sqrt{a^{3}} + 2 \sqrt{a}}{2 \sqrt{a}}$

Step 1.2

Factor $2$ out of $- 8 \sqrt{a^{3}}$ .

$\frac{2 (8 \sqrt{a^{5}}) + 2 (- 4 \sqrt{a^{3}}) + 2 \sqrt{a}}{2 \sqrt{a}}$

Step 1.3

Factor $2$ out of $2 (8 \sqrt{a^{5}}) + 2 (- 4 \sqrt{a^{3}})$ .

$\frac{2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}}) + 2 \sqrt{a}}{2 \sqrt{a}}$

Step 1.4

Factor $2$ out of $2 \sqrt{a}$ .

$\frac{2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}}) + 2 (\sqrt{a})}{2 \sqrt{a}}$

Step 1.5

Factor $2$ out of $2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}}) + 2 (\sqrt{a})$ .

$\frac{2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}} + \sqrt{a})}{2 \sqrt{a}}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $2$ out of $2 \sqrt{a}$ .

$\frac{2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}} + \sqrt{a})}{2 (\sqrt{a})}$

Step 1.6.2

Cancel the common factor.

$\frac{2 (8 \sqrt{a^{5}} - 4 \sqrt{a^{3}} + \sqrt{a})}{2 \sqrt{a}}$

Step 1.6.3

Rewrite the expression.

$\frac{8 \sqrt{a^{5}} - 4 \sqrt{a^{3}} + \sqrt{a}}{\sqrt{a}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{8 a^{\frac{5}{2}} - 4 \sqrt{a^{3}} + \sqrt{a}}{\sqrt{a}}$

Step 2.2

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

$\frac{8 a^{\frac{5}{2}} - 4 a^{\frac{3}{2}} + \sqrt{a}}{\sqrt{a}}$

Step 2.3

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

$\frac{8 a^{\frac{5}{2}} - 4 a^{\frac{3}{2}} + a^{\frac{1}{2}}}{\sqrt{a}}$

Step 2.4

Factor $a^{\frac{1}{2}}$ out of $8 a^{\frac{5}{2}} - 4 a^{\frac{3}{2}} + a^{\frac{1}{2}}$ .

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Step 2.4.1

Factor $a^{\frac{1}{2}}$ out of $8 a^{\frac{5}{2}}$ .

$\frac{a^{\frac{1}{2}} (8 a^{\frac{4}{2}}) - 4 a^{\frac{3}{2}} + a^{\frac{1}{2}}}{\sqrt{a}}$

Step 2.4.2

Factor $a^{\frac{1}{2}}$ out of $- 4 a^{\frac{3}{2}}$ .

$\frac{a^{\frac{1}{2}} (8 a^{\frac{4}{2}}) + a^{\frac{1}{2}} (- 4 a^{\frac{2}{2}}) + a^{\frac{1}{2}}}{\sqrt{a}}$

Step 2.4.3

Multiply by $1$ .

$\frac{a^{\frac{1}{2}} (8 a^{\frac{4}{2}}) + a^{\frac{1}{2}} (- 4 a^{\frac{2}{2}}) + a^{\frac{1}{2}} \cdot 1}{\sqrt{a}}$

Step 2.4.4

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (8 a^{\frac{4}{2}}) + a^{\frac{1}{2}} (- 4 a^{\frac{2}{2}})$ .

$\frac{a^{\frac{1}{2}} (8 a^{\frac{4}{2}} - 4 a^{\frac{2}{2}}) + a^{\frac{1}{2}} \cdot 1}{\sqrt{a}}$

Step 2.4.5

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (8 a^{\frac{4}{2}} - 4 a^{\frac{2}{2}}) + a^{\frac{1}{2}} \cdot 1$ .

$\frac{a^{\frac{1}{2}} (8 a^{\frac{4}{2}} - 4 a^{\frac{2}{2}} + 1)}{\sqrt{a}}$

Step 2.5

Divide $4$ by $2$ .

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a^{\frac{2}{2}} + 1)}{\sqrt{a}}$

Step 2.6

Divide $2$ by $2$ .

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a^{1} + 1)}{\sqrt{a}}$

Step 2.7

Simplify.

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1)}{\sqrt{a}}$

Step 3

Multiply $\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1)}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1)}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1)}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{\sqrt{a} \sqrt{a}}$

Step 4.2

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

Step 4.3

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

Step 4.4

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{{\sqrt{a}}^{2}}$

Step 4.6

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

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Step 4.6.1

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{a^{1}}$

Step 4.6.5

Simplify.

$\frac{a^{\frac{1}{2}} (8 a^{2} - 4 a + 1) \sqrt{a}}{a}$

Step 5

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

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a \cdot a^{- \frac{1}{2}}}$

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{1} a^{- \frac{1}{2}}}$

Step 6.1.2

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

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{1 - \frac{1}{2}}}$

Step 6.2

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

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{2}{2} - \frac{1}{2}}}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{2 - 1}{2}}}$

Step 6.4

Subtract $1$ from $2$ .

$\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{1}{2}}}$

Step 7

Reorder factors in $\frac{(8 a^{2} - 4 a + 1) \sqrt{a}}{a^{\frac{1}{2}}}$ .

$\frac{\sqrt{a} (8 a^{2} - 4 a + 1)}{a^{\frac{1}{2}}}$