Algebra Examples

$\frac{- 2 f}{d^{5}} {(\frac{d f^{5}}{- 2 f^{10}})}^{2}$

Step 1

Move the negative in front of the fraction.

$- \frac{2 f}{d^{5}} \cdot {(\frac{d f^{5}}{- 2 f^{10}})}^{2}$

Step 2

Cancel the common factor of $f^{5}$ and $f^{10}$ .

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Step 2.1

Factor $f^{5}$ out of $d f^{5}$ .

$- \frac{2 f}{d^{5}} \cdot {(\frac{f^{5} d}{- 2 f^{10}})}^{2}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $f^{5}$ out of $- 2 f^{10}$ .

$- \frac{2 f}{d^{5}} \cdot {(\frac{f^{5} d}{f^{5} (- 2 f^{5})})}^{2}$

Step 2.2.2

Cancel the common factor.

$- \frac{2 f}{d^{5}} \cdot {(\frac{f^{5} d}{f^{5} (- 2 f^{5})})}^{2}$

Step 2.2.3

Rewrite the expression.

$- \frac{2 f}{d^{5}} \cdot {(\frac{d}{- 2 f^{5}})}^{2}$

Step 3

Move the negative in front of the fraction.

$- \frac{2 f}{d^{5}} \cdot {(- \frac{d}{2 f^{5}})}^{2}$

Step 4

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

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Step 4.1

Apply the product rule to $- \frac{d}{2 f^{5}}$ .

$- \frac{2 f}{d^{5}} \cdot ({(- 1)}^{2} {(\frac{d}{2 f^{5}})}^{2})$

Step 4.2

Apply the product rule to $\frac{d}{2 f^{5}}$ .

$- \frac{2 f}{d^{5}} \cdot ({(- 1)}^{2} \frac{d^{2}}{{(2 f^{5})}^{2}})$

Step 4.3

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

$- \frac{2 f}{d^{5}} \cdot ({(- 1)}^{2} \frac{d^{2}}{2^{2} {(f^{5})}^{2}})$

Step 5

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 5.1

Move ${(- 1)}^{2}$ .

${(- 1)}^{2} \cdot - 1 \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{2^{2} {(f^{5})}^{2}}$

Step 5.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 5.2.1

Raise $- 1$ to the power of $1$ .

${(- 1)}^{2} \cdot {(- 1)}^{1} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{2^{2} {(f^{5})}^{2}}$

Step 5.2.2

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

${(- 1)}^{2 + 1} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{2^{2} {(f^{5})}^{2}}$

Step 5.3

Add $2$ and $1$ .

${(- 1)}^{3} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{2^{2} {(f^{5})}^{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

${(- 1)}^{3} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{4 {(f^{5})}^{2}}$

Step 6.2

Multiply the exponents in ${(f^{5})}^{2}$ .

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Step 6.2.1

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

${(- 1)}^{3} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{4 f^{5 \cdot 2}}$

Step 6.2.2

Multiply $5$ by $2$ .

${(- 1)}^{3} \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{4 f^{10}}$

Step 7

Simplify terms.

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Step 7.1

Raise $- 1$ to the power of $3$ .

$- \frac{2 f}{d^{5}} \cdot \frac{d^{2}}{4 f^{10}}$

Step 7.2

Cancel the common factor of $2 f$ .

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Step 7.2.1

Move the leading negative in $- \frac{2 f}{d^{5}}$ into the numerator.

$\frac{- 2 f}{d^{5}} \cdot \frac{d^{2}}{4 f^{10}}$

Step 7.2.2

Factor $2 f$ out of $- 2 f$ .

$\frac{2 f (- 1)}{d^{5}} \cdot \frac{d^{2}}{4 f^{10}}$

Step 7.2.3

Factor $2 f$ out of $4 f^{10}$ .

$\frac{2 f (- 1)}{d^{5}} \cdot \frac{d^{2}}{2 f (2 f^{9})}$

Step 7.2.4

Cancel the common factor.

$\frac{2 f \cdot - 1}{d^{5}} \cdot \frac{d^{2}}{2 f (2 f^{9})}$

Step 7.2.5

Rewrite the expression.

$\frac{- 1}{d^{5}} \cdot \frac{d^{2}}{2 f^{9}}$

Step 7.3

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

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Step 7.3.1

Factor $d^{2}$ out of $d^{5}$ .

$\frac{- 1}{d^{2} d^{3}} \cdot \frac{d^{2}}{2 f^{9}}$

Step 7.3.2

Cancel the common factor.

$\frac{- 1}{d^{2} d^{3}} \cdot \frac{d^{2}}{2 f^{9}}$

Step 7.3.3

Rewrite the expression.

$\frac{- 1}{d^{3}} \cdot \frac{1}{2 f^{9}}$

Step 7.4

Multiply $\frac{- 1}{d^{3}}$ by $\frac{1}{2 f^{9}}$ .

$\frac{- 1}{d^{3} (2 f^{9})}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Move $2$ to the left of $d^{3}$ .

$\frac{- 1}{2 \cdot d^{3} f^{9}}$

Step 7.5.2

Move the negative in front of the fraction.

$- \frac{1}{2 d^{3} f^{9}}$