Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 a^{2}}$ as ${(8 a^{2})}^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 1.3

Apply the product rule to $8 a^{2}$ .

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

Step 1.4

Multiply the exponents in ${(a^{2})}^{\frac{1}{2}}$ .

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.5

Simplify.

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

Step 1.6

Apply the product rule to $2 a$ .

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

Step 1.7

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

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Step 1.7.1

Factor $a^{\frac{1}{2}}$ out of $3 \cdot 8^{\frac{1}{2}} a$ .

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

Step 1.7.2

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

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

Step 1.7.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} a^{\frac{1}{2}}) + a^{\frac{1}{2}} (- 2^{\frac{1}{2}})$ .

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Add $- 1$ and $2$ .

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

Step 6

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

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