Algebra Examples

$\frac{{(4 k^{3} m^{2})}^{3}}{{(5 k^{2} m^{- 3})}^{- 2}}$

Step 1

Move ${(5 k^{2} m^{- 3})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(4 k^{3} m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 2

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

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Step 2.1

Apply the product rule to $4 k^{3} m^{2}$ .

${(4 k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 2.2

Apply the product rule to $4 k^{3}$ .

$4^{3} {(k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 3

Raise $4$ to the power of $3$ .

$64 {(k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 4

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

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Step 4.1

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

$64 k^{3 \cdot 3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 4.2

Multiply $3$ by $3$ .

$64 k^{9} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Step 5

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

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Step 5.1

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

$64 k^{9} m^{2 \cdot 3} {(5 k^{2} m^{- 3})}^{2}$

Step 5.2

Multiply $2$ by $3$ .

$64 k^{9} m^{6} {(5 k^{2} m^{- 3})}^{2}$

Step 6

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

$64 k^{9} m^{6} {(5 k^{2} \frac{1}{m^{3}})}^{2}$

Step 7

Multiply $5 k^{2} \frac{1}{m^{3}}$ .

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Step 7.1

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

$64 k^{9} m^{6} {(k^{2} \frac{5}{m^{3}})}^{2}$

Step 7.2

Combine $k^{2}$ and $\frac{5}{m^{3}}$ .

$64 k^{9} m^{6} {(\frac{k^{2} \cdot 5}{m^{3}})}^{2}$

Step 8

Move $5$ to the left of $k^{2}$ .

$64 k^{9} m^{6} {(\frac{5 \cdot k^{2}}{m^{3}})}^{2}$

Step 9

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

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Step 9.1

Apply the product rule to $\frac{5 k^{2}}{m^{3}}$ .

$64 k^{9} m^{6} \frac{{(5 k^{2})}^{2}}{{(m^{3})}^{2}}$

Step 9.2

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

$64 k^{9} m^{6} \frac{5^{2} {(k^{2})}^{2}}{{(m^{3})}^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

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

$64 k^{9} m^{6} \frac{25 {(k^{2})}^{2}}{{(m^{3})}^{2}}$

Step 10.2

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

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Step 10.2.1

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

$64 k^{9} m^{6} \frac{25 k^{2 \cdot 2}}{{(m^{3})}^{2}}$

Step 10.2.2

Multiply $2$ by $2$ .

$64 k^{9} m^{6} \frac{25 k^{4}}{{(m^{3})}^{2}}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

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

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Step 11.1.1

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

$64 k^{9} m^{6} \frac{25 k^{4}}{m^{3 \cdot 2}}$

Step 11.1.2

Multiply $3$ by $2$ .

$64 k^{9} m^{6} \frac{25 k^{4}}{m^{6}}$

Step 11.2

Cancel the common factor of $m^{6}$ .

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Step 11.2.1

Factor $m^{6}$ out of $64 k^{9} m^{6}$ .

$m^{6} (64 k^{9}) \frac{25 k^{4}}{m^{6}}$

Step 11.2.2

Cancel the common factor.

$m^{6} (64 k^{9}) \frac{25 k^{4}}{m^{6}}$

Step 11.2.3

Rewrite the expression.

$64 k^{9} (25 k^{4})$

Step 11.3

Multiply $25$ by $64$ .

$1600 k^{9} k^{4}$

Step 12

Multiply $k^{9}$ by $k^{4}$ by adding the exponents.

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Step 12.1

Move $k^{4}$ .

$1600 (k^{4} k^{9})$

Step 12.2

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

$1600 k^{4 + 9}$

Step 12.3

Add $4$ and $9$ .

$1600 k^{13}$