Simplify (2qr^5)^3(qr)^6

${(2 q r^{5})}^{3} {(q r)}^{6}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 q r^{5}$ .

${(2 q)}^{3} {(r^{5})}^{3} {(q r)}^{6}$

Apply the product rule to $2 q$ .

$2^{3} q^{3} {(r^{5})}^{3} {(q r)}^{6}$

Step 2

Raise $2$ to the power of $3$ .

$8 q^{3} {(r^{5})}^{3} {(q r)}^{6}$

Step 3

Multiply the exponents in ${(r^{5})}^{3}$ .

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

$8 q^{3} r^{5 \cdot 3} {(q r)}^{6}$

Multiply $5$ by $3$ .

$8 q^{3} r^{15} {(q r)}^{6}$

Step 4

Apply the product rule to $q r$ .

$8 q^{3} r^{15} (q^{6} r^{6})$

Step 5

Multiply $q^{3}$ by $q^{6}$ by adding the exponents.

Move $q^{6}$ .

$8 (q^{6} q^{3}) r^{15} r^{6}$

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

$8 q^{6 + 3} r^{15} r^{6}$

Add $6$ and $3$ .

$8 q^{9} r^{15} r^{6}$

Step 6

Multiply $r^{15}$ by $r^{6}$ by adding the exponents.

Move $r^{6}$ .

$8 q^{9} (r^{6} r^{15})$

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

$8 q^{9} r^{6 + 15}$

Add $6$ and $15$ .

$8 q^{9} r^{21}$