Basic Math Examples

${(2 a^{3} b^{4})}^{- 3} {(4 a^{2} b)}^{2}$

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $2 a^{3} b^{4}$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{1}{8 {(a^{3})}^{3} {(b^{4})}^{3}} {(4 a^{2} b)}^{2}$

Step 2.4

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

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Step 2.4.1

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

$\frac{1}{8 a^{3 \cdot 3} {(b^{4})}^{3}} {(4 a^{2} b)}^{2}$

Step 2.4.2

Multiply $3$ by $3$ .

$\frac{1}{8 a^{9} {(b^{4})}^{3}} {(4 a^{2} b)}^{2}$

Step 2.5

Multiply the exponents in ${(b^{4})}^{3}$ .

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Step 2.5.1

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

$\frac{1}{8 a^{9} b^{4 \cdot 3}} {(4 a^{2} b)}^{2}$

Step 2.5.2

Multiply $4$ by $3$ .

$\frac{1}{8 a^{9} b^{12}} {(4 a^{2} b)}^{2}$

Step 3

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

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Step 3.1

Apply the product rule to $4 a^{2} b$ .

$\frac{1}{8 a^{9} b^{12}} ({(4 a^{2})}^{2} b^{2})$

Step 3.2

Apply the product rule to $4 a^{2}$ .

$\frac{1}{8 a^{9} b^{12}} (4^{2} {(a^{2})}^{2} b^{2})$

Step 4

Simplify the expression.

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Step 4.1

Rewrite using the commutative property of multiplication.

$4^{2} \frac{1}{8 a^{9} b^{12}} ({(a^{2})}^{2} b^{2})$

Step 4.2

Raise $4$ to the power of $2$ .

$16 \frac{1}{8 a^{9} b^{12}} ({(a^{2})}^{2} b^{2})$

Step 5

Cancel the common factor of $8$ .

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Step 5.1

Factor $8$ out of $16$ .

$8 (2) \frac{1}{8 a^{9} b^{12}} ({(a^{2})}^{2} b^{2})$

Step 5.2

Factor $8$ out of $8 a^{9} b^{12}$ .

$8 (2) \frac{1}{8 (a^{9} b^{12})} ({(a^{2})}^{2} b^{2})$

Step 5.3

Cancel the common factor.

$8 \cdot 2 \frac{1}{8 (a^{9} b^{12})} ({(a^{2})}^{2} b^{2})$

Step 5.4

Rewrite the expression.

$2 \frac{1}{a^{9} b^{12}} ({(a^{2})}^{2} b^{2})$

Step 6

Combine $2$ and $\frac{1}{a^{9} b^{12}}$ .

$\frac{2}{a^{9} b^{12}} ({(a^{2})}^{2} b^{2})$

Step 7

Cancel the common factor of $b^{2}$ .

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Step 7.1

Factor $b^{2}$ out of $a^{9} b^{12}$ .

$\frac{2}{b^{2} (a^{9} b^{10})} ({(a^{2})}^{2} b^{2})$

Step 7.2

Factor $b^{2}$ out of ${(a^{2})}^{2} b^{2}$ .

$\frac{2}{b^{2} (a^{9} b^{10})} (b^{2} {(a^{2})}^{2})$

Step 7.3

Cancel the common factor.

$\frac{2}{b^{2} (a^{9} b^{10})} (b^{2} {(a^{2})}^{2})$

Step 7.4

Rewrite the expression.

$\frac{2}{a^{9} b^{10}} {(a^{2})}^{2}$

Step 8

Combine $\frac{2}{a^{9} b^{10}}$ and ${(a^{2})}^{2}$ .

$\frac{2 {(a^{2})}^{2}}{a^{9} b^{10}}$

Step 9

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

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Step 9.1

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

$\frac{2 a^{2 \cdot 2}}{a^{9} b^{10}}$

Step 9.2

Multiply $2$ by $2$ .

$\frac{2 a^{4}}{a^{9} b^{10}}$

Step 10

Cancel the common factor of $a^{4}$ and $a^{9}$ .

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Step 10.1

Factor $a^{4}$ out of $2 a^{4}$ .

$\frac{a^{4} \cdot 2}{a^{9} b^{10}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $a^{4}$ out of $a^{9} b^{10}$ .

$\frac{a^{4} \cdot 2}{a^{4} (a^{5} b^{10})}$

Step 10.2.2

Cancel the common factor.

$\frac{a^{4} \cdot 2}{a^{4} (a^{5} b^{10})}$

Step 10.2.3

Rewrite the expression.

$\frac{2}{a^{5} b^{10}}$