Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\frac{1+i}{i}-\frac{3}{4-i}$$

Step-by-Step Solution

Answer

The result in standard form is \( -5/17 - (20i/17) \).

Step 1 - Simplify the first fraction

For the first fraction, we can simply divide \(1+i\) by \(i\). Remember that \(i\) divided by \(i\) is 1, and 1 divided by \(i\) is \(-i\). Hence, the fraction simplifies to \(1+i\div i\)= \(1-i\).

Step 2 - Simplify the second fraction using conjugate

To simplify the second fraction \(3/(4-i)\), we will need to remove the \(i\) from the denominator. For this, we can multiply and divide by the conjugate of the denominator, which is \(4+i\). On multiplying with the conjugate, the denominator becomes \((4-i) *(4+i)\) = \(4^2 - i^2\) = \(16 - (-1)\) = \(17\) (since \(i^2\) = -1 by definition). So now, the second term becomes \(3*(4+i) / 17 = (12+3i)/17\).

Step 3 - Perform the subtraction

Since we have simplified both terms of the expression, it's now time to subtract them: \((1-i) - (12+3i)/17 = 1- 12/17 -i -3i/17 = -5/17 - (20i/17)\).