Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $- 2 a^{5} b^{3}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $5$ by $3$ .

$2 a^{3} \cdot \frac{1}{- 8 a^{15} {(b^{3})}^{3}}$

Step 2.5

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

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Step 2.5.1

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

$2 a^{3} \cdot \frac{1}{- 8 a^{15} b^{3 \cdot 3}}$

Step 2.5.2

Multiply $3$ by $3$ .

$2 a^{3} \cdot \frac{1}{- 8 a^{15} b^{9}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $2 a^{3}$ .

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Step 3.1.1

Factor $2 a^{3}$ out of $- 8 a^{15} b^{9}$ .

$2 a^{3} \cdot \frac{1}{2 a^{3} (- 4 a^{12} b^{9})}$

Step 3.1.2

Cancel the common factor.

$2 a^{3} \cdot \frac{1}{2 a^{3} (- 4 a^{12} b^{9})}$

Step 3.1.3

Rewrite the expression.

$\frac{1}{- 4 a^{12} b^{9}}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{4 a^{12} b^{9}}$