Basic Math Examples

$5 \sqrt{18 c^{3}} \cdot \sqrt{3 c^{5}}$

Step 1

Rewrite $18 c^{3}$ as ${(3 c)}^{2} \cdot (2 c)$ .

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Step 1.1

Factor $9$ out of $18$ .

$5 \sqrt{9 (2) c^{3}} \cdot \sqrt{3 c^{5}}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$5 \sqrt{3^{2} \cdot 2 c^{3}} \cdot \sqrt{3 c^{5}}$

Step 1.3

Factor out $c^{2}$ .

$5 \sqrt{3^{2} \cdot 2 (c^{2} c)} \cdot \sqrt{3 c^{5}}$

Step 1.4

Move $2$ .

$5 \sqrt{3^{2} c^{2} \cdot 2 c} \cdot \sqrt{3 c^{5}}$

Step 1.5

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

$5 \sqrt{{(3 c)}^{2} \cdot 2 c} \cdot \sqrt{3 c^{5}}$

Step 1.6

Add parentheses.

$5 \sqrt{{(3 c)}^{2} \cdot (2 c)} \cdot \sqrt{3 c^{5}}$

Step 2

Pull terms out from under the radical.

$5 (3 c \sqrt{2 c}) \cdot \sqrt{3 c^{5}}$

Step 3

Multiply $3$ by $5$ .

$15 (c \sqrt{2 c}) \cdot \sqrt{3 c^{5}}$

Step 4

Rewrite $3 c^{5}$ as ${(c^{2})}^{2} \cdot (3 c)$ .

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Step 4.1

Factor out $c^{4}$ .

$15 c \sqrt{2 c} \cdot \sqrt{3 (c^{4} c)}$

Step 4.2

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

$15 c \sqrt{2 c} \cdot \sqrt{3 ({(c^{2})}^{2} c)}$

Step 4.3

Reorder $3$ and ${(c^{2})}^{2}$ .

$15 c \sqrt{2 c} \cdot \sqrt{{(c^{2})}^{2} \cdot 3 c}$

Step 4.4

Add parentheses.

$15 c \sqrt{2 c} \cdot \sqrt{{(c^{2})}^{2} \cdot (3 c)}$

Step 5

Pull terms out from under the radical.

$15 c \sqrt{2 c} \cdot (c^{2} \sqrt{3 c})$

Step 6

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

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Step 6.1

Move $c^{2}$ .

$15 (c^{2} c) \sqrt{2 c} \cdot \sqrt{3 c}$

Step 6.2

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

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Step 6.2.1

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

$15 (c^{2} c^{1}) \sqrt{2 c} \cdot \sqrt{3 c}$

Step 6.2.2

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

$15 c^{2 + 1} \sqrt{2 c} \cdot \sqrt{3 c}$

Step 6.3

Add $2$ and $1$ .

$15 c^{3} \sqrt{2 c} \cdot \sqrt{3 c}$

Step 7

Multiply $15 c^{3} \sqrt{2 c} \sqrt{3 c}$ .

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Step 7.1

Combine using the product rule for radicals.

$15 c^{3} \sqrt{3 c (2 c)}$

Step 7.2

Multiply $2$ by $3$ .

$15 c^{3} \sqrt{6 c \cdot c}$

Step 7.3

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

$15 c^{3} \sqrt{6 (c^{1} c)}$

Step 7.4

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

$15 c^{3} \sqrt{6 (c^{1} c^{1})}$

Step 7.5

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

$15 c^{3} \sqrt{6 c^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$15 c^{3} \sqrt{6 c^{2}}$

Step 8

Reorder $6$ and $c^{2}$ .

$15 c^{3} \sqrt{c^{2} \cdot 6}$

Step 9

Pull terms out from under the radical.

$15 c^{3} (c \sqrt{6})$

Step 10

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

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Step 10.1

Move $c$ .

$15 (c \cdot c^{3}) \sqrt{6}$

Step 10.2

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

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Step 10.2.1

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

$15 (c^{1} c^{3}) \sqrt{6}$

Step 10.2.2

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

$15 c^{1 + 3} \sqrt{6}$

Step 10.3

Add $1$ and $3$ .

$15 c^{4} \sqrt{6}$