Simplify the Radical Expression square root of (r^4g^6p^2)^3

$\sqrt{{(r^{4} g^{6} p^{2})}^{3}}$

Step 1

Apply the product rule to $r^{4} g^{6} p^{2}$ .

$\sqrt{{(r^{4} g^{6})}^{3} {(p^{2})}^{3}}$

Step 2

Apply the product rule to $r^{4} g^{6}$ .

$\sqrt{{(r^{4})}^{3} {(g^{6})}^{3} {(p^{2})}^{3}}$

Step 3

Multiply the exponents in ${(r^{4})}^{3}$ .

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

$\sqrt{r^{4 \cdot 3} {(g^{6})}^{3} {(p^{2})}^{3}}$

Multiply $4$ by $3$ .

$\sqrt{r^{12} {(g^{6})}^{3} {(p^{2})}^{3}}$

Step 4

Multiply the exponents in ${(g^{6})}^{3}$ .

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

$\sqrt{r^{12} g^{6 \cdot 3} {(p^{2})}^{3}}$

Multiply $6$ by $3$ .

$\sqrt{r^{12} g^{18} {(p^{2})}^{3}}$

Step 5

Multiply the exponents in ${(p^{2})}^{3}$ .

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

$\sqrt{r^{12} g^{18} p^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\sqrt{r^{12} g^{18} p^{6}}$

Step 6

Rewrite $r^{12} g^{18} p^{6}$ as ${(r^{6} g^{9} p^{3})}^{2}$ .

$\sqrt{{(r^{6} g^{9} p^{3})}^{2}}$

Step 7

Pull terms out from under the radical.

$| r^{6} g^{9} p^{3} |$

Step 8

Remove non-negative terms from the absolute value.

$r^{6} | g^{9} p^{3} |$