Simplify (-2a^6bc^3)^2*-5ab^2

${(- 2 a^{6} b c^{3})}^{2} \cdot - 5 a b^{2}$

Step 1

Rewrite using the commutative property of multiplication.

$- 5 {(- 2 a^{6} b c^{3})}^{2} a b^{2}$

Step 2

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

Apply the product rule to $- 2 a^{6} b c^{3}$ .

$- 5 ({(- 2 a^{6} b)}^{2} {(c^{3})}^{2}) a b^{2}$

Apply the product rule to $- 2 a^{6} b$ .

$- 5 ({(- 2 a^{6})}^{2} b^{2} {(c^{3})}^{2}) a b^{2}$

Apply the product rule to $- 2 a^{6}$ .

$- 5 ({(- 2)}^{2} {(a^{6})}^{2} b^{2} {(c^{3})}^{2}) a b^{2}$

Step 3

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$- 5 ({(- 2)}^{2} {(a^{6})}^{2} (b^{2} b^{2}) {(c^{3})}^{2}) a$

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

$- 5 ({(- 2)}^{2} {(a^{6})}^{2} b^{2 + 2} {(c^{3})}^{2}) a$

Add $2$ and $2$ .

$- 5 ({(- 2)}^{2} {(a^{6})}^{2} b^{4} {(c^{3})}^{2}) a$

Step 4

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

$- 5 (4 {(a^{6})}^{2} b^{4} {(c^{3})}^{2}) a$

Step 5

Multiply the exponents in ${(a^{6})}^{2}$ .

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

$- 5 (4 a^{6 \cdot 2} b^{4} {(c^{3})}^{2}) a$

Multiply $6$ by $2$ .

$- 5 (4 a^{12} b^{4} {(c^{3})}^{2}) a$

Step 6

Multiply $a^{12}$ by $a$ by adding the exponents.

Move $a$ .

$- 5 (4 (a \cdot a^{12}) b^{4} {(c^{3})}^{2})$

Multiply $a$ by $a^{12}$ .

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

$- 5 (4 (a^{1} a^{12}) b^{4} {(c^{3})}^{2})$

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

$- 5 (4 a^{1 + 12} b^{4} {(c^{3})}^{2})$

Add $1$ and $12$ .

$- 5 (4 a^{13} b^{4} {(c^{3})}^{2})$

Step 7

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

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

$- 5 (4 a^{13} b^{4} c^{3 \cdot 2})$

Multiply $3$ by $2$ .

$- 5 (4 a^{13} b^{4} c^{6})$

Step 8

Multiply $4$ by $- 5$ .

$- 20 a^{13} b^{4} c^{6}$

Simplify (-2a^6bc^3)^2*-5ab^2

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