Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $- 3 a^{2} b^{- 1}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $2$ .

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

Step 1.5

Multiply the exponents in ${(b^{- 1})}^{2}$ .

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Step 1.5.1

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

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

Step 1.5.2

Multiply $- 1$ by $2$ .

$\frac{9 a^{4} b^{- 2}}{3 a b^{2}}$

Step 1.6

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

$\frac{9 a^{4} \frac{1}{b^{2}}}{3 a b^{2}}$

Step 1.7

Combine exponents.

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Step 1.7.1

Combine $9$ and $\frac{1}{b^{2}}$ .

$\frac{a^{4} \frac{9}{b^{2}}}{3 a b^{2}}$

Step 1.7.2

Combine $a^{4}$ and $\frac{9}{b^{2}}$ .

$\frac{\frac{a^{4} \cdot 9}{b^{2}}}{3 a b^{2}}$

Step 1.8

Move $9$ to the left of $a^{4}$ .

$\frac{\frac{9 a^{4}}{b^{2}}}{3 a b^{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 a^{4}}{b^{2}} \cdot \frac{1}{3 a b^{2}}$

Step 3

Combine.

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

Step 4

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

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Step 4.1

Move $b^{2}$ .

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

Step 4.2

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

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

Step 4.3

Add $2$ and $2$ .

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

Step 5

Multiply $9 a^{4}$ by $1$ .

$\frac{9 a^{4}}{b^{4} (3 a)}$

Step 6

Cancel the common factor of $9$ and $3$ .

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Step 6.1

Factor $3$ out of $9 a^{4}$ .

$\frac{3 (3 a^{4})}{b^{4} (3 a)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $b^{4} (3 a)$ .

$\frac{3 (3 a^{4})}{3 (b^{4} (a))}$

Step 6.2.2

Cancel the common factor.

$\frac{3 (3 a^{4})}{3 (b^{4} (a))}$

Step 6.2.3

Rewrite the expression.

$\frac{3 a^{4}}{b^{4} (a)}$

Step 7

Cancel the common factor of $a^{4}$ and $a$ .

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Step 7.1

Factor $a$ out of $3 a^{4}$ .

$\frac{a (3 a^{3})}{b^{4} (a)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $a$ out of $b^{4} (a)$ .

$\frac{a (3 a^{3})}{a b^{4}}$

Step 7.2.2

Cancel the common factor.

$\frac{a (3 a^{3})}{a b^{4}}$

Step 7.2.3

Rewrite the expression.

$\frac{3 a^{3}}{b^{4}}$