Basic Math Examples

$v^{2} (\frac{6}{\sqrt{12 v - 1}}) + \sqrt{12 v - 1} (2 v)$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{6}{\sqrt{12 v - 1}}$ by $\frac{\sqrt{12 v - 1}}{\sqrt{12 v - 1}}$ .

$v^{2} (\frac{6}{\sqrt{12 v - 1}} \cdot \frac{\sqrt{12 v - 1}}{\sqrt{12 v - 1}}) + \sqrt{12 v - 1} (2 v)$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{6}{\sqrt{12 v - 1}}$ by $\frac{\sqrt{12 v - 1}}{\sqrt{12 v - 1}}$ .

$v^{2} \frac{6 \sqrt{12 v - 1}}{\sqrt{12 v - 1} \sqrt{12 v - 1}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.2

Raise $\sqrt{12 v - 1}$ to the power of $1$ .

$v^{2} \frac{6 \sqrt{12 v - 1}}{{\sqrt{12 v - 1}}^{1} \sqrt{12 v - 1}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.3

Raise $\sqrt{12 v - 1}$ to the power of $1$ .

$v^{2} \frac{6 \sqrt{12 v - 1}}{{\sqrt{12 v - 1}}^{1} {\sqrt{12 v - 1}}^{1}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.4

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

$v^{2} \frac{6 \sqrt{12 v - 1}}{{\sqrt{12 v - 1}}^{1 + 1}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.5

Add $1$ and $1$ .

$v^{2} \frac{6 \sqrt{12 v - 1}}{{\sqrt{12 v - 1}}^{2}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.6

Rewrite ${\sqrt{12 v - 1}}^{2}$ as $12 v - 1$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 v - 1}$ as ${(12 v - 1)}^{\frac{1}{2}}$ .

$v^{2} \frac{6 \sqrt{12 v - 1}}{{({(12 v - 1)}^{\frac{1}{2}})}^{2}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.6.2

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

$v^{2} \frac{6 \sqrt{12 v - 1}}{{(12 v - 1)}^{\frac{1}{2} \cdot 2}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.6.3

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

$v^{2} \frac{6 \sqrt{12 v - 1}}{{(12 v - 1)}^{\frac{2}{2}}} + \sqrt{12 v - 1} (2 v)$

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.

$v^{2} \frac{6 \sqrt{12 v - 1}}{{(12 v - 1)}^{\frac{2}{2}}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.6.4.2

Rewrite the expression.

$v^{2} \frac{6 \sqrt{12 v - 1}}{{(12 v - 1)}^{1}} + \sqrt{12 v - 1} (2 v)$

Step 1.2.6.5

Simplify.

$v^{2} \frac{6 \sqrt{12 v - 1}}{12 v - 1} + \sqrt{12 v - 1} (2 v)$

Step 1.3

Combine $v^{2}$ and $\frac{6 \sqrt{12 v - 1}}{12 v - 1}$ .

$\frac{v^{2} (6 \sqrt{12 v - 1})}{12 v - 1} + \sqrt{12 v - 1} (2 v)$

Step 1.4

Move $6$ to the left of $v^{2}$ .

$\frac{6 \cdot v^{2} \sqrt{12 v - 1}}{12 v - 1} + \sqrt{12 v - 1} (2 v)$

Step 1.5

Rewrite using the commutative property of multiplication.

$\frac{6 v^{2} \sqrt{12 v - 1}}{12 v - 1} + 2 \sqrt{12 v - 1} v$

Step 2

To write $2 \sqrt{12 v - 1} v$ as a fraction with a common denominator, multiply by $\frac{12 v - 1}{12 v - 1}$ .

$\frac{6 v^{2} \sqrt{12 v - 1}}{12 v - 1} + \frac{2 \sqrt{12 v - 1} v (12 v - 1)}{12 v - 1}$

Step 3

Combine the numerators over the common denominator.

$\frac{6 v^{2} \sqrt{12 v - 1} + 2 \sqrt{12 v - 1} v (12 v - 1)}{12 v - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $2 v \sqrt{12 v - 1}$ out of $6 v^{2} \sqrt{12 v - 1} + 2 \sqrt{12 v - 1} v (12 v - 1)$ .

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Step 4.1.1

Factor $2 v \sqrt{12 v - 1}$ out of $6 v^{2} \sqrt{12 v - 1}$ .

$\frac{2 v \sqrt{12 v - 1} (3 v) + 2 \sqrt{12 v - 1} v (12 v - 1)}{12 v - 1}$

Step 4.1.2

Factor $2 v \sqrt{12 v - 1}$ out of $2 \sqrt{12 v - 1} v (12 v - 1)$ .

$\frac{2 v \sqrt{12 v - 1} (3 v) + 2 v \sqrt{12 v - 1} (12 v - 1)}{12 v - 1}$

Step 4.1.3

Factor $2 v \sqrt{12 v - 1}$ out of $2 v \sqrt{12 v - 1} (3 v) + 2 v \sqrt{12 v - 1} (12 v - 1)$ .

$\frac{2 v \sqrt{12 v - 1} (3 v + 12 v - 1)}{12 v - 1}$

Step 4.2

Add $3 v$ and $12 v$ .

$\frac{2 v \sqrt{12 v - 1} (15 v - 1)}{12 v - 1}$