Basic Math Examples

$2 \sqrt{c d^{6} e^{3}} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 1

Rewrite $c d^{6} e^{3}$ as ${(d^{3} e)}^{2} (c e)$ .

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Step 1.1

Rewrite $d^{6}$ as ${(d^{3})}^{2}$ .

$2 \sqrt{c {(d^{3})}^{2} e^{3}} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 1.2

Factor out $e^{2}$ .

$2 \sqrt{c {(d^{3})}^{2} (e^{2} e)} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 1.3

Move $c$ .

$2 \sqrt{{(d^{3})}^{2} e^{2} c e} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 1.4

Rewrite ${(d^{3})}^{2} e^{2}$ as ${(d^{3} e)}^{2}$ .

$2 \sqrt{{(d^{3} e)}^{2} c e} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 1.5

Add parentheses.

$2 \sqrt{{(d^{3} e)}^{2} (c e)} \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 2

Pull terms out from under the radical.

$2 (d^{3} e \sqrt{c e}) \cdot \sqrt{c^{2} d^{3} e^{4}}$

Step 3

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

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Step 3.1

Factor out $d^{2}$ .

$2 d^{3} e \sqrt{c e} \cdot \sqrt{c^{2} (d^{2} d) e^{4}}$

Step 3.2

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

$2 d^{3} e \sqrt{c e} \cdot \sqrt{c^{2} (d^{2} d) {(e^{2})}^{2}}$

Step 3.3

Move $d$ .

$2 d^{3} e \sqrt{c e} \cdot \sqrt{c^{2} (d^{2}) {(e^{2})}^{2} d}$

Step 3.4

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

$2 d^{3} e \sqrt{c e} \cdot \sqrt{{(c d e^{2})}^{2} d}$

Step 4

Pull terms out from under the radical.

$2 d^{3} e \sqrt{c e} \cdot (c d e^{2} \sqrt{d})$

Step 5

Multiply $d^{3}$ by $d$ by adding the exponents.

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Step 5.1

Move $d$ .

$2 (d \cdot d^{3}) e \sqrt{c e} \cdot (c e^{2} \sqrt{d})$

Step 5.2

Multiply $d$ by $d^{3}$ .

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Step 5.2.1

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

$2 (d^{1} d^{3}) e \sqrt{c e} \cdot (c e^{2} \sqrt{d})$

Step 5.2.2

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

$2 d^{1 + 3} e \sqrt{c e} \cdot (c e^{2} \sqrt{d})$

Step 5.3

Add $1$ and $3$ .

$2 d^{4} e \sqrt{c e} \cdot (c e^{2} \sqrt{d})$

Step 6

Multiply $e$ by $e^{2}$ by adding the exponents.

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Step 6.1

Move $e^{2}$ .

$2 d^{4} (e^{2} e) \sqrt{c e} \cdot (c \sqrt{d})$

Step 6.2

Multiply $e^{2}$ by $e$ .

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Step 6.2.1

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

$2 d^{4} (e^{2} e^{1}) \sqrt{c e} \cdot (c \sqrt{d})$

Step 6.2.2

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

$2 d^{4} e^{2 + 1} \sqrt{c e} \cdot (c \sqrt{d})$

Step 6.3

Add $2$ and $1$ .

$2 d^{4} e^{3} \sqrt{c e} \cdot (c \sqrt{d})$

Step 7

Combine using the product rule for radicals.

$2 d^{4} e^{3} (c \sqrt{c e d})$

Step 8

Reorder factors in $2 d^{4} e^{3} (c \sqrt{c e d})$ .

$2 d^{4} e^{3} c \sqrt{c e d}$