Simplify (-2x^5)^3(-5xy^6)^2

${(- 2 x^{5})}^{3} {(- 5 x y^{6})}^{2}$

Step 1

Apply the product rule to $- 2 x^{5}$ .

${(- 2)}^{3} {(x^{5})}^{3} {(- 5 x y^{6})}^{2}$

Step 2

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

$- 8 {(x^{5})}^{3} {(- 5 x y^{6})}^{2}$

Step 3

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

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

$- 8 x^{5 \cdot 3} {(- 5 x y^{6})}^{2}$

Multiply $5$ by $3$ .

$- 8 x^{15} {(- 5 x y^{6})}^{2}$

Step 4

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

Apply the product rule to $- 5 x y^{6}$ .

$- 8 x^{15} ({(- 5 x)}^{2} {(y^{6})}^{2})$

Apply the product rule to $- 5 x$ .

$- 8 x^{15} ({(- 5)}^{2} x^{2} {(y^{6})}^{2})$

Step 5

Multiply $x^{15}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$- 8 (x^{2} x^{15}) ({(- 5)}^{2} {(y^{6})}^{2})$

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

$- 8 x^{2 + 15} ({(- 5)}^{2} {(y^{6})}^{2})$

Add $2$ and $15$ .

$- 8 x^{17} ({(- 5)}^{2} {(y^{6})}^{2})$

Step 6

Rewrite using the commutative property of multiplication.

$- 8 \cdot {(- 5)}^{2} x^{17} {(y^{6})}^{2}$

Step 7

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

$- 8 \cdot 25 x^{17} {(y^{6})}^{2}$

Step 8

Multiply $- 8$ by $25$ .

$- 200 x^{17} {(y^{6})}^{2}$

Step 9

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

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

$- 200 x^{17} y^{6 \cdot 2}$

Multiply $6$ by $2$ .

$- 200 x^{17} y^{12}$