Reduce (1/1-4i -2/1+ i)(3-4i/5 +i)to the standard form?

Question

Answer

1/(1 -4i) - 2/(1+i)}(3-4i)/(5+i)= {(1+i-2(1-4i)}/(1-4i)(1+i) × (3-4i)/(5+i) = (9i -1)(3-4i)/(1-4i)(1+i)(5+i)= (-3 +4i +27i -36i^2)/{1(1+I)-4i(1+i)}(5+i)= {-3 + 31i -36(-1)}/(1+i-4i-4i^2)(5+i)= {33 +31i}/(1-3i+4)(5+i)= (33+31i)/(5-3i)(5+i)= (33+ 31i)/(25 +5i -15i -3i^2)= (33 + 31i)/(25 -10i +3) = (33+31i)(28 +10i)/(28-10i)(28+10i)= {33(28+10i)+31i(28+10i)}/(28^2+10^2)= {924 + 330i + 868i +310i^2}/884= {614 + 1198i}/884 = (307 + 599i)/442 Hence, (307/442) + (599/442)i

Answer

Answer

5 +2i

Answer

Solution:

To reduce the given expression to the standard form, we need to simplify the expression by performing the necessary operations. Let's break down the expression into smaller parts and simplify each part step by step.

Part 1: Simplifying (1/1 - 4i)

To simplify (1/1 - 4i), we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (1 + 4i).

So, we have:
(1/1 - 4i) = (1/1 - 4i) * (1 + 4i) / (1 + 4i)
= (1 * (1 + 4i) + (-4i) * (1 + 4i)) / (1 + 4i)
= (1 + 4i - 4i - 16i^2) / (1 + 4i)
= (1 + 4i - 4i - 16(-1)) / (1 + 4i)
= (1 + 4i - 4i + 16) / (1 + 4i)
= (17 + 0i) / (1 + 4i)
= 17 / (1 + 4i)

Part 2: Simplifying (-2/1 i)

To simplify (-2/1 i), we can rewrite it as (-2i).

Part 3: Simplifying (3 - 4i/5 i)

To simplify (3 - 4i/5i), we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (-5i).

So, we have:
(3 - 4i/5i) = (3 - 4i/5i) * (-5i) / (-5i)
= (3 * (-5i) - 4i * (-5i)) / (-5i)
= (-15i + 20i^2) / (-5i)
= (-15i + 20(-1)) / (-5i)
= (-15i - 20) / (-5i)
= (-20 - 15i) / (-5i)
= 4 + 3i

Part 4: Combining the Simplified Expressions

Now, we can substitute the simplified expressions back into the original expression and simplify further.

(1/1 - 4i - 2/1 i)(3 - 4i/5 i) = (17 / (1 + 4i)) * (4 + 3i)
= (17 * 4 + 17 * 3i) / (1 + 4i)
= (68 + 51i) / (1 + 4i)

We can further simplify this expression by rationalizing the denominator using the conjugate of the denominator, which is (1 - 4i).

(68 + 51i) / (1 + 4i) = ((68 + 51

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