Simplify (-2a^-2)^3(-5a^-3)^2

${(- 2 a^{- 2})}^{3} {(- 5 a^{- 3})}^{2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(- 2 \frac{1}{a^{2}})}^{3} {(- 5 a^{- 3})}^{2}$

Step 2

Combine $- 2$ and $\frac{1}{a^{2}}$ .

${(\frac{- 2}{a^{2}})}^{3} {(- 5 a^{- 3})}^{2}$

Step 3

Move the negative in front of the fraction.

${(- \frac{2}{a^{2}})}^{3} {(- 5 a^{- 3})}^{2}$

Step 4

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

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

${(- 1)}^{3} {(\frac{2}{a^{2}})}^{3} {(- 5 a^{- 3})}^{2}$

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

${(- 1)}^{3} \frac{2^{3}}{{(a^{2})}^{3}} {(- 5 a^{- 3})}^{2}$

Step 5

Raise $- 1$ to the power of $3$ .

$- \frac{2^{3}}{{(a^{2})}^{3}} {(- 5 a^{- 3})}^{2}$

Step 6

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

$- \frac{8}{{(a^{2})}^{3}} {(- 5 a^{- 3})}^{2}$

Step 7

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

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

$- \frac{8}{a^{2 \cdot 3}} {(- 5 a^{- 3})}^{2}$

Multiply $2$ by $3$ .

$- \frac{8}{a^{6}} {(- 5 a^{- 3})}^{2}$

Step 8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{8}{a^{6}} {(- 5 \frac{1}{a^{3}})}^{2}$

Step 9

Combine $- 5$ and $\frac{1}{a^{3}}$ .

$- \frac{8}{a^{6}} {(\frac{- 5}{a^{3}})}^{2}$

Step 10

Move the negative in front of the fraction.

$- \frac{8}{a^{6}} {(- \frac{5}{a^{3}})}^{2}$

Step 11

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

Apply the product rule to $- \frac{5}{a^{3}}$ .

$- \frac{8}{a^{6}} ({(- 1)}^{2} {(\frac{5}{a^{3}})}^{2})$

Apply the product rule to $\frac{5}{a^{3}}$ .

$- \frac{8}{a^{6}} ({(- 1)}^{2} \frac{5^{2}}{{(a^{3})}^{2}})$

Step 12

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

${(- 1)}^{2} \cdot - 1 \frac{8}{a^{6}} \frac{5^{2}}{{(a^{3})}^{2}}$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

${(- 1)}^{2} \cdot {(- 1)}^{1} \frac{8}{a^{6}} \frac{5^{2}}{{(a^{3})}^{2}}$

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

${(- 1)}^{2 + 1} \frac{8}{a^{6}} \frac{5^{2}}{{(a^{3})}^{2}}$

Add $2$ and $1$ .

${(- 1)}^{3} \frac{8}{a^{6}} \frac{5^{2}}{{(a^{3})}^{2}}$

Step 13

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

${(- 1)}^{3} \frac{8}{a^{6}} \frac{25}{{(a^{3})}^{2}}$

Step 14

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

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

${(- 1)}^{3} \frac{8}{a^{6}} \frac{25}{a^{3 \cdot 2}}$

Multiply $3$ by $2$ .

${(- 1)}^{3} \frac{8}{a^{6}} \frac{25}{a^{6}}$

Step 15

Raise $- 1$ to the power of $3$ .

$- \frac{8}{a^{6}} \cdot \frac{25}{a^{6}}$

Step 16

Multiply $- \frac{8}{a^{6}} \cdot \frac{25}{a^{6}}$ .

Multiply $\frac{25}{a^{6}}$ by $\frac{8}{a^{6}}$ .

$- \frac{25 \cdot 8}{a^{6} a^{6}}$

Multiply $25$ by $8$ .

$- \frac{200}{a^{6} a^{6}}$

Multiply $a^{6}$ by $a^{6}$ by adding the exponents.

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

$- \frac{200}{a^{6 + 6}}$

Add $6$ and $6$ .

$- \frac{200}{a^{12}}$

Simplify (-2a^-2)^3(-5a^-3)^2

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