Algebra Examples

$\frac{{(5 m^{4} k^{- 1})}^{- 4}}{2 m^{- 3} k^{- 9}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(5 m^{4} k^{- 1})}^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 m^{- 3} k^{- 9} {(5 m^{4} k^{- 1})}^{4}}$

Step 1.2

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

$\frac{m^{3}}{2 k^{- 9} {(5 m^{4} k^{- 1})}^{4}}$

Step 1.3

Move $k^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{m^{3} k^{9}}{2 {(5 m^{4} k^{- 1})}^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $5 m^{4} k^{- 1}$ .

$\frac{m^{3} k^{9}}{2 {(5 m^{4})}^{4} {(k^{- 1})}^{4}}$

Step 2.2

Apply the product rule to $5 m^{4}$ .

$\frac{m^{3} k^{9}}{2 (5^{4} {(m^{4})}^{4}) {(k^{- 1})}^{4}}$

Step 2.3

Raise $5$ to the power of $4$ .

$\frac{m^{3} k^{9}}{2 (625 {(m^{4})}^{4}) {(k^{- 1})}^{4}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{m^{3} k^{9}}{2 (625 m^{4 \cdot 4}) {(k^{- 1})}^{4}}$

Step 2.4.2

Multiply $4$ by $4$ .

$\frac{m^{3} k^{9}}{2 (625 m^{16}) {(k^{- 1})}^{4}}$

Step 2.5

Multiply the exponents in ${(k^{- 1})}^{4}$ .

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Step 2.5.1

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

$\frac{m^{3} k^{9}}{2 (625 m^{16}) k^{- 1 \cdot 4}}$

Step 2.5.2

Multiply $- 1$ by $4$ .

$\frac{m^{3} k^{9}}{2 (625 m^{16}) k^{- 4}}$

Step 2.6

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

$\frac{m^{3} k^{9}}{2 \cdot 625 m^{16} \frac{1}{k^{4}}}$

Step 2.7

Combine exponents.

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Step 2.7.1

Multiply $2$ by $625$ .

$\frac{m^{3} k^{9}}{1250 m^{16} \frac{1}{k^{4}}}$

Step 2.7.2

Combine $1250$ and $\frac{1}{k^{4}}$ .

$\frac{m^{3} k^{9}}{m^{16} \frac{1250}{k^{4}}}$

Step 2.7.3

Combine $m^{16}$ and $\frac{1250}{k^{4}}$ .

$\frac{m^{3} k^{9}}{\frac{m^{16} \cdot 1250}{k^{4}}}$

Step 2.8

Move $1250$ to the left of $m^{16}$ .

$\frac{m^{3} k^{9}}{\frac{1250 m^{16}}{k^{4}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$m^{3} k^{9} \frac{k^{4}}{1250 m^{16}}$

Step 4

Cancel the common factor of $m^{3}$ .

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Step 4.1

Factor $m^{3}$ out of $m^{3} k^{9}$ .

$m^{3} (k^{9}) \frac{k^{4}}{1250 m^{16}}$

Step 4.2

Factor $m^{3}$ out of $1250 m^{16}$ .

$m^{3} (k^{9}) \frac{k^{4}}{m^{3} (1250 m^{13})}$

Step 4.3

Cancel the common factor.

$m^{3} k^{9} \frac{k^{4}}{m^{3} (1250 m^{13})}$

Step 4.4

Rewrite the expression.

$k^{9} \frac{k^{4}}{1250 m^{13}}$

Step 5

Combine $k^{9}$ and $\frac{k^{4}}{1250 m^{13}}$ .

$\frac{k^{9} k^{4}}{1250 m^{13}}$

Step 6

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

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Step 6.1

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

$\frac{k^{9 + 4}}{1250 m^{13}}$

Step 6.2

Add $9$ and $4$ .

$\frac{k^{13}}{1250 m^{13}}$