Algebra Examples

$\frac{\sqrt{32 c^{5} d^{3}}}{\sqrt{2 a b}}$

Step 1

Combine $\sqrt{32 c^{5} d^{3}}$ and $\sqrt{2 a b}$ into a single radical.

$\sqrt{\frac{32 c^{5} d^{3}}{2 a b}}$

Step 2

Reduce the expression $\frac{32 c^{5} d^{3}}{2 a b}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $32 c^{5} d^{3}$ .

$\sqrt{\frac{2 (16 c^{5} d^{3})}{2 a b}}$

Step 2.2

Factor $2$ out of $2 a b$ .

$\sqrt{\frac{2 (16 c^{5} d^{3})}{2 (a b)}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{2 (16 c^{5} d^{3})}{2 (a b)}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{16 c^{5} d^{3}}{a b}}$

Step 3

Rewrite $\sqrt{\frac{16 c^{5} d^{3}}{a b}}$ as $\frac{\sqrt{16 c^{5} d^{3}}}{\sqrt{a b}}$ .

$\frac{\sqrt{16 c^{5} d^{3}}}{\sqrt{a b}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $16 c^{5} d^{3}$ as ${(4 c^{2} d)}^{2} (c d)$ .

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Step 4.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} c^{5} d^{3}}}{\sqrt{a b}}$

Step 4.1.2

Factor out $c^{4}$ .

$\frac{\sqrt{4^{2} (c^{4} c) d^{3}}}{\sqrt{a b}}$

Step 4.1.3

Rewrite $c^{4}$ as ${(c^{2})}^{2}$ .

$\frac{\sqrt{4^{2} ({(c^{2})}^{2} c) d^{3}}}{\sqrt{a b}}$

Step 4.1.4

Factor out $d^{2}$ .

$\frac{\sqrt{4^{2} ({(c^{2})}^{2} c) (d^{2} d)}}{\sqrt{a b}}$

Step 4.1.5

Move $c$ .

$\frac{\sqrt{4^{2} ({(c^{2})}^{2}) d^{2} c d}}{\sqrt{a b}}$

Step 4.1.6

Rewrite $4^{2} ({(c^{2})}^{2}) d^{2}$ as ${(4 c^{2} d)}^{2}$ .

$\frac{\sqrt{{(4 c^{2} d)}^{2} c d}}{\sqrt{a b}}$

Step 4.1.7

Add parentheses.

$\frac{\sqrt{{(4 c^{2} d)}^{2} (c d)}}{\sqrt{a b}}$

Step 4.2

Pull terms out from under the radical.

$\frac{4 c^{2} d \sqrt{c d}}{\sqrt{a b}}$

Step 5

Multiply $\frac{4 c^{2} d \sqrt{c d}}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

$\frac{4 c^{2} d \sqrt{c d}}{\sqrt{a b}} \cdot \frac{\sqrt{a b}}{\sqrt{a b}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{4 c^{2} d \sqrt{c d}}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{\sqrt{a b} \sqrt{a b}}$

Step 6.2

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{\sqrt{a b}}^{1} \sqrt{a b}}$

Step 6.3

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{\sqrt{a b}}^{1} {\sqrt{a b}}^{1}}$

Step 6.4

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{\sqrt{a b}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{\sqrt{a b}}^{2}}$

Step 6.6

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

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Step 6.6.1

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{({(a b)}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{(a b)}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{(a b)}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{(a b)}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{{(a b)}^{1}}$

Step 6.6.5

Simplify.

$\frac{4 c^{2} d \sqrt{c d} \sqrt{a b}}{a b}$

Step 7

Combine using the product rule for radicals.

$\frac{4 c^{2} d \sqrt{a b c d}}{a b}$