Simplify (5x^3y)^2(-2x^5y^1)

${(5 x^{3} y)}^{2} (- 2 x^{5} y^{1})$

Step 1

Rewrite using the commutative property of multiplication.

$- 2 {(5 x^{3} y)}^{2} x^{5} y^{1}$

Step 2

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

Apply the product rule to $5 x^{3} y$ .

$- 2 ({(5 x^{3})}^{2} y^{2}) x^{5} y^{1}$

Apply the product rule to $5 x^{3}$ .

$- 2 (5^{2} {(x^{3})}^{2} y^{2}) x^{5} y^{1}$

Step 3

Multiply $y^{2}$ by $y^{1}$ by adding the exponents.

Move $y^{1}$ .

$- 2 (5^{2} {(x^{3})}^{2} (y^{1} y^{2})) x^{5}$

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

$- 2 (5^{2} {(x^{3})}^{2} y^{1 + 2}) x^{5}$

Add $1$ and $2$ .

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

Step 4

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

$- 2 (25 {(x^{3})}^{2} y^{3}) x^{5}$

Step 5

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

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

$- 2 (25 x^{3 \cdot 2} y^{3}) x^{5}$

Multiply $3$ by $2$ .

$- 2 (25 x^{6} y^{3}) x^{5}$

Step 6

Multiply $x^{6}$ by $x^{5}$ by adding the exponents.

Move $x^{5}$ .

$- 2 (25 (x^{5} x^{6}) y^{3})$

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

$- 2 (25 x^{5 + 6} y^{3})$

Add $5$ and $6$ .

$- 2 (25 x^{11} y^{3})$

Step 7

Multiply $25$ by $- 2$ .

$- 50 x^{11} y^{3}$