Simplify (-2f)/(d^5)((df^5)/(-2f^10))^2

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

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 $f^{10}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

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.

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

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

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

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

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.

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

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

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

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

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

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

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

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.

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

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

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

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

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

Multiply $5$ by $2$ .

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

Step 7

Simplify terms.

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

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

Cancel the common factor of $2 f$ .

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Simplify (-2f)/(d^5)((df^5)/(-2f^10))^2

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