Solve 20/5j/frac{1{20j}+1/-5+1/-10j}

$$\frac { \frac { 20 } { 5 j } } { \frac { 1 } { 20 j } + \frac { 1 } { - 5 } + \frac { 1 } { - 10 j } }$$

Evaluate

$$-\frac{80}{4j+1}$$

Short Solution Steps

Fraction \frac{1}{-5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$\frac{\frac{20}{5j}}{\frac{1}{20j}-\frac{1}{5}+\frac{1}{-10j}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20j and 5 is 20j. Multiply \frac{1}{5} times \frac{4j}{4j}.

$$\frac{\frac{20}{5j}}{\frac{1}{20j}-\frac{4j}{20j}+\frac{1}{-10j}}$$

Since \frac{1}{20j} and \frac{4j}{20j} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{20}{5j}}{\frac{1-4j}{20j}+\frac{1}{-10j}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20j and -10j is 20j. Multiply \frac{1}{-10j} times \frac{-2}{-2}.

$$\frac{\frac{20}{5j}}{\frac{1-4j}{20j}+\frac{-2}{20j}}$$

Since \frac{1-4j}{20j} and \frac{-2}{20j} have the same denominator, add them by adding their numerators.

$$\frac{\frac{20}{5j}}{\frac{1-4j-2}{20j}}$$

Combine like terms in 1-4j-2.

$$\frac{\frac{20}{5j}}{\frac{-1-4j}{20j}}$$

Divide \frac{20}{5j} by \frac{-1-4j}{20j} by multiplying \frac{20}{5j} by the reciprocal of \frac{-1-4j}{20j}.

$$\frac{20\times 20j}{5j\left(-1-4j\right)}$$

Cancel out 5j in both numerator and denominator.

$$\frac{4\times 20}{-4j-1}$$

Multiply 4 and 20 to get 80.

$$\frac{80}{-4j-1}$$

Differentiate w.r.t. j

$$\frac{320}{\left(4j+1\right)^{2}}$$