Show the value of the expression 25 ( (1+j3)/(3+j4) - (2-j5)/(j) )

To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $3+j4$ is $3-j4$.

$$\frac{1+j3}{3+j4} = \frac{1+j3}{3+j4} \times \frac{3-j4}{3-j4}$$

Multiply the numerators:

$$(1+j3)(3-j4) = 1(3) + 1(-j4) + j3(3) + j3(-j4)$$ $$= 3 - j4 + j9 - j^212$$

Since $j^2 = -1$:

$$= 3 + j5 - (-1)12 = 3 + j5 + 12 = 15 + j5$$

Multiply the denominators:

$$(3+j4)(3-j4) = 3^2 - (j4)^2 = 9 - j^216$$ $$= 9 - (-1)16 = 9 + 16 = 25$$

So, the first fraction simplifies to:

$$\frac{15+j5}{25} = \frac{15}{25} + \frac{j5}{25} = \frac{3}{5} + j\frac{1}{5}$$

To simplify this fraction, multiply the numerator and denominator by $-j$.

$$\frac{2-j5}{j} = \frac{2-j5}{j} \times \frac{-j}{-j}$$

Multiply the numerators:

$$(2-j5)(-j) = -j2 + j^25$$

Since $j^2 = -1$:

$$= -j2 + (-1)5 = -5 - j2$$

Multiply the denominators:

$$j(-j) = -j^2 = -(-1) = 1$$

So, the second fraction simplifies to:

$$\frac{-5-j2}{1} = -5 - j2$$

$$\left( \frac{3}{5} + j\frac{1}{5} \right) - (-5 - j2)$$ $$= \frac{3}{5} + j\frac{1}{5} + 5 + j2$$

Combine the real parts:

$$\frac{3}{5} + 5 = \frac{3}{5} + \frac{25}{5} = \frac{28}{5}$$

Combine the imaginary parts:

$$j\frac{1}{5} + j2 = j\frac{1}{5} + j\frac{10}{5} = j\frac{11}{5}$$

So, the expression inside the parenthesis is:

$$\frac{28}{5} + j\frac{11}{5}$$

$$25 \left( \frac{28}{5} + j\frac{11}{5} \right)$$ $$= 25 \times \frac{28}{5} + 25 \times j\frac{11}{5}$$ $$= 5 \times 28 + 5 \times j11$$ $$= 140 + j55$$

The simplified expression is:

$$\mathbf{140 + j55}$$