Algebra Examples

$\sqrt{v^{19}}$

Step 1

Multiply $\sqrt{v^{19}}$ by $1$ .

$\sqrt{v^{19}}$

Step 2

Multiply to rationalize the numerator.

$\frac{\sqrt{v^{19}} \sqrt{v^{19}}}{\sqrt{v^{19}}}$

Step 3

Simplify.

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Step 3.1

Raise $\sqrt{v^{19}}$ to the power of $1$ .

$\frac{{\sqrt{v^{19}}}^{1} \sqrt{v^{19}}}{\sqrt{v^{19}}}$

Step 3.2

Raise $\sqrt{v^{19}}$ to the power of $1$ .

$\frac{{\sqrt{v^{19}}}^{1} {\sqrt{v^{19}}}^{1}}{\sqrt{v^{19}}}$

Step 3.3

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

$\frac{{\sqrt{v^{19}}}^{1 + 1}}{\sqrt{v^{19}}}$

Step 3.4

Add $1$ and $1$ .

$\frac{{\sqrt{v^{19}}}^{2}}{\sqrt{v^{19}}}$

Step 3.5

Rewrite ${\sqrt{v^{19}}}^{2}$ as $v^{19}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v^{19}}$ as $v^{\frac{19}{2}}$ .

$\frac{{(v^{\frac{19}{2}})}^{2}}{\sqrt{v^{19}}}$

Step 3.5.2

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

$\frac{v^{\frac{19}{2} \cdot 2}}{\sqrt{v^{19}}}$

Step 3.5.3

Combine $\frac{19}{2}$ and $2$ .

$\frac{v^{\frac{19 \cdot 2}{2}}}{\sqrt{v^{19}}}$

Step 3.5.4

Multiply $19$ by $2$ .

$\frac{v^{\frac{38}{2}}}{\sqrt{v^{19}}}$

Step 3.5.5

Cancel the common factor of $38$ and $2$ .

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Step 3.5.5.1

Factor $2$ out of $38$ .

$\frac{v^{\frac{2 \cdot 19}{2}}}{\sqrt{v^{19}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $2$ out of $2$ .

$\frac{v^{\frac{2 \cdot 19}{2 (1)}}}{\sqrt{v^{19}}}$

Step 3.5.5.2.2

Cancel the common factor.

$\frac{v^{\frac{2 \cdot 19}{2 \cdot 1}}}{\sqrt{v^{19}}}$

Step 3.5.5.2.3

Rewrite the expression.

$\frac{v^{\frac{19}{1}}}{\sqrt{v^{19}}}$

Step 3.5.5.2.4

Divide $19$ by $1$ .

$\frac{v^{19}}{\sqrt{v^{19}}}$

Step 4

Rewrite $v^{19}$ as ${(v^{9})}^{2} v$ .

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Step 4.1

Factor out $v^{18}$ .

$\frac{v^{19}}{\sqrt{v^{18} v}}$

Step 4.2

Rewrite $v^{18}$ as ${(v^{9})}^{2}$ .

$\frac{v^{19}}{\sqrt{{(v^{9})}^{2} v}}$

Step 5

Pull terms out from under the radical.

$\frac{v^{19}}{v^{9} \sqrt{v}}$