Simplify 3 square root of 5c* square root of 15c^3

$3 \sqrt{5 c} \cdot \sqrt{15 c^{3}}$

Step 1

Rewrite $15 c^{3}$ as $c^{2} \cdot (15 c)$ .

Factor out $c^{2}$ .

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

Reorder $15$ and $c^{2}$ .

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

Add parentheses.

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $3 \sqrt{5 c} (c \sqrt{15 c})$ .

Combine using the product rule for radicals.

$3 (c \sqrt{5 c (15 c)})$

Multiply $15$ by $5$ .

$3 (c \sqrt{75 c \cdot c})$

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

$3 (c \sqrt{75 (c^{1} c)})$

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

$3 (c \sqrt{75 (c^{1} c^{1})})$

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

$3 (c \sqrt{75 c^{1 + 1}})$

Add $1$ and $1$ .

$3 (c \sqrt{75 c^{2}})$

$3 (c \sqrt{75 c^{2}})$

Step 4

Rewrite $75 c^{2}$ as ${(5 c)}^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$3 (c \sqrt{25 (3) c^{2}})$

Rewrite $25$ as $5^{2}$ .

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

Move $3$ .

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

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

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

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

Step 5

Pull terms out from under the radical.

$3 (c (5 c \sqrt{3}))$

Step 6

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

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

Multiply $c$ by $c$ .

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

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

Step 7

Move $5$ to the left of $c^{2}$ .

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

Step 8

Multiply $5$ by $3$ .

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

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