Algebra Examples

$\frac{10 k^{- 10} b^{- 9}}{5 {(k^{- 5} b^{2})}^{- 3}}$

Step 1

Move $k^{- 10}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10 b^{- 9}}{5 {(k^{- 5} b^{2})}^{- 3} k^{10}}$

Step 2

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

$\frac{10}{5 {(k^{- 5} b^{2})}^{- 3} k^{10} b^{9}}$

Step 3

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

$\frac{10 {(k^{- 5} b^{2})}^{3}}{5 k^{10} b^{9}}$

Step 4

Cancel the common factor of $10$ and $5$ .

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Step 4.1

Factor $5$ out of $10 {(k^{- 5} b^{2})}^{3}$ .

$\frac{5 (2 {(k^{- 5} b^{2})}^{3})}{5 k^{10} b^{9}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $5 k^{10} b^{9}$ .

$\frac{5 (2 {(k^{- 5} b^{2})}^{3})}{5 (k^{10} b^{9})}$

Step 4.2.2

Cancel the common factor.

$\frac{5 (2 {(k^{- 5} b^{2})}^{3})}{5 (k^{10} b^{9})}$

Step 4.2.3

Rewrite the expression.

$\frac{2 {(k^{- 5} b^{2})}^{3}}{k^{10} b^{9}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{2 {(k^{- 5})}^{3} {(b^{2})}^{3}}{k^{10} b^{9}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{2 k^{- 5 \cdot 3} {(b^{2})}^{3}}{k^{10} b^{9}}$

Step 5.2.2

Multiply $- 5$ by $3$ .

$\frac{2 k^{- 15} {(b^{2})}^{3}}{k^{10} b^{9}}$

Step 5.3

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

$\frac{2 \frac{1}{k^{15}} {(b^{2})}^{3}}{k^{10} b^{9}}$

Step 5.4

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

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Step 5.4.1

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

$\frac{2 \frac{1}{k^{15}} b^{2 \cdot 3}}{k^{10} b^{9}}$

Step 5.4.2

Multiply $2$ by $3$ .

$\frac{2 \frac{1}{k^{15}} b^{6}}{k^{10} b^{9}}$

Step 5.5

Combine exponents.

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Step 5.5.1

Combine $2$ and $\frac{1}{k^{15}}$ .

$\frac{\frac{2}{k^{15}} b^{6}}{k^{10} b^{9}}$

Step 5.5.2

Combine $\frac{2}{k^{15}}$ and $b^{6}$ .

$\frac{\frac{2 b^{6}}{k^{15}}}{k^{10} b^{9}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 b^{6}}{k^{15}} \cdot \frac{1}{k^{10} b^{9}}$

Step 7

Combine.

$\frac{2 b^{6} \cdot 1}{k^{15} (k^{10} b^{9})}$

Step 8

Multiply $k^{15}$ by $k^{10}$ by adding the exponents.

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Step 8.1

Move $k^{10}$ .

$\frac{2 b^{6} \cdot 1}{k^{10} k^{15} b^{9}}$

Step 8.2

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

$\frac{2 b^{6} \cdot 1}{k^{10 + 15} b^{9}}$

Step 8.3

Add $10$ and $15$ .

$\frac{2 b^{6} \cdot 1}{k^{25} b^{9}}$

Step 9

Multiply $2 b^{6}$ by $1$ .

$\frac{2 b^{6}}{k^{25} b^{9}}$

Step 10

Cancel the common factor of $b^{6}$ and $b^{9}$ .

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Step 10.1

Factor $b^{6}$ out of $2 b^{6}$ .

$\frac{b^{6} \cdot 2}{k^{25} b^{9}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $b^{6}$ out of $k^{25} b^{9}$ .

$\frac{b^{6} \cdot 2}{b^{6} (k^{25} b^{3})}$

Step 10.2.2

Cancel the common factor.

$\frac{b^{6} \cdot 2}{b^{6} (k^{25} b^{3})}$

Step 10.2.3

Rewrite the expression.

$\frac{2}{k^{25} b^{3}}$