Simplify (3pq)^4*(p^6q)-(2p^2q)^5

${(3 p q)}^{4} \cdot (p^{6} q) - {(2 p^{2} q)}^{5}$

Step 1

Simplify each term.

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

Apply the product rule to $3 p q$ .

${(3 p)}^{4} q^{4} \cdot (p^{6} q) - {(2 p^{2} q)}^{5}$

Apply the product rule to $3 p$ .

$3^{4} p^{4} q^{4} \cdot (p^{6} q) - {(2 p^{2} q)}^{5}$

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

Move $p^{6}$ .

$3^{4} (p^{6} p^{4}) q^{4} \cdot q - {(2 p^{2} q)}^{5}$

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

$3^{4} p^{6 + 4} q^{4} \cdot q - {(2 p^{2} q)}^{5}$

Add $6$ and $4$ .

$3^{4} p^{10} q^{4} \cdot q - {(2 p^{2} q)}^{5}$

Multiply $q^{4}$ by $q$ by adding the exponents.

Move $q$ .

$3^{4} p^{10} (q \cdot q^{4}) - {(2 p^{2} q)}^{5}$

Multiply $q$ by $q^{4}$ .

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

$3^{4} p^{10} (q^{1} q^{4}) - {(2 p^{2} q)}^{5}$

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

$3^{4} p^{10} q^{1 + 4} - {(2 p^{2} q)}^{5}$

Add $1$ and $4$ .

$3^{4} p^{10} q^{5} - {(2 p^{2} q)}^{5}$

Raise $3$ to the power of $4$ .

$81 p^{10} q^{5} - {(2 p^{2} q)}^{5}$

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

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

$81 p^{10} q^{5} - ({(2 p^{2})}^{5} q^{5})$

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

$81 p^{10} q^{5} - (2^{5} {(p^{2})}^{5} q^{5})$

Raise $2$ to the power of $5$ .

$81 p^{10} q^{5} - (32 {(p^{2})}^{5} q^{5})$

Multiply the exponents in ${(p^{2})}^{5}$ .

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

$81 p^{10} q^{5} - (32 p^{2 \cdot 5} q^{5})$

Multiply $2$ by $5$ .

$81 p^{10} q^{5} - (32 p^{10} q^{5})$

Multiply $32$ by $- 1$ .

$81 p^{10} q^{5} - 32 p^{10} q^{5}$

Step 2

Subtract $32 p^{10} q^{5}$ from $81 p^{10} q^{5}$ .

$49 p^{10} q^{5}$