Simplify (1/2p^2q)^3(2pq^2)^4

${(\frac{1}{2} p^{2} q)}^{3} {(2 p q^{2})}^{4}$

Step 1

Multiply $\frac{1}{2} (p^{2} q)$ .

Combine $p^{2}$ and $\frac{1}{2}$ .

${(\frac{p^{2}}{2} q)}^{3} {(2 p q^{2})}^{4}$

Combine $\frac{p^{2}}{2}$ and $q$ .

${(\frac{p^{2} q}{2})}^{3} {(2 p q^{2})}^{4}$

Step 2

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

Apply the product rule to $\frac{p^{2} q}{2}$ .

$\frac{{(p^{2} q)}^{3}}{2^{3}} {(2 p q^{2})}^{4}$

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

$\frac{{(p^{2})}^{3} q^{3}}{2^{3}} {(2 p q^{2})}^{4}$

Step 3

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

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

$\frac{p^{2 \cdot 3} q^{3}}{2^{3}} {(2 p q^{2})}^{4}$

Multiply $2$ by $3$ .

$\frac{p^{6} q^{3}}{2^{3}} {(2 p q^{2})}^{4}$

Step 4

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

$\frac{p^{6} q^{3}}{8} {(2 p q^{2})}^{4}$

Step 5

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

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

$\frac{p^{6} q^{3}}{8} ({(2 p)}^{4} {(q^{2})}^{4})$

Apply the product rule to $2 p$ .

$\frac{p^{6} q^{3}}{8} (2^{4} p^{4} {(q^{2})}^{4})$

Step 6

Simplify the expression.

Rewrite using the commutative property of multiplication.

$2^{4} \frac{p^{6} q^{3}}{8} (p^{4} {(q^{2})}^{4})$

Raise $2$ to the power of $4$ .

$16 \frac{p^{6} q^{3}}{8} (p^{4} {(q^{2})}^{4})$

Step 7

Cancel the common factor of $8$ .

Factor $8$ out of $16$ .

$8 (2) \frac{p^{6} q^{3}}{8} (p^{4} {(q^{2})}^{4})$

Cancel the common factor.

$8 \cdot 2 \frac{p^{6} q^{3}}{8} (p^{4} {(q^{2})}^{4})$

Rewrite the expression.

$2 (p^{6} q^{3}) (p^{4} {(q^{2})}^{4})$

Step 8

Multiply $p^{6}$ by $p^{4}$ by adding the exponents.

Move $p^{4}$ .

$2 (p^{4} p^{6} q^{3}) {(q^{2})}^{4}$

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

$2 (p^{4 + 6} q^{3}) {(q^{2})}^{4}$

Add $4$ and $6$ .

$2 (p^{10} q^{3}) {(q^{2})}^{4}$

Step 9

Multiply the exponents in ${(q^{2})}^{4}$ .

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

$2 p^{10} q^{3} q^{2 \cdot 4}$

Multiply $2$ by $4$ .

$2 p^{10} q^{3} q^{8}$

Step 10

Multiply $q^{3}$ by $q^{8}$ by adding the exponents.

Move $q^{8}$ .

$2 p^{10} (q^{8} q^{3})$

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

$2 p^{10} q^{8 + 3}$

Add $8$ and $3$ .

$2 p^{10} q^{11}$

Simplify (1/2p^2q)^3(2pq^2)^4

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