SOLUTION: I'm not quite sure how to simplify the expression 4 over 2+5i (4/2+5i) and I'm not quite sure what i to the -35th power means.

Question 147558: I'm not quite sure how to simplify the expression 4 over 2+5i (4/2+5i) and I'm not quite sure what i to the -35th power means.
Found 3 solutions by jim_thompson5910, stanbon, Nate:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Start with the given expression.

Multiply both numerator and denominator by the complex conjugate of

Combine the fractions

Foil the denominator to get

Distribute

So

------------------------------

Start with the given expression.

Flip the fraction to make the exponent positive.

Now let's find evaluate :

First take the exponent 35 and divide by 4.
When it leaves a remainder of 0, the answer is .
When it leaves a remainder of 1, the answer is .
When it leaves a remainder of 2, the answer is .
When it leaves a remainder of 3, the answer is .

Since leaves a remainder of 3, this means the answer is .

So

So this means that

Start with the given expression.

Multiply both numerator and denominator by "i"

Combine the fractions.

Multiply and to get

Replace with . Remember

Multiply

Simplify

So
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
I'm not quite sure how to simplify the expression 4 over 2+5i (4/2+5i) and I'm not quite sure what i to the -35th power means.
----------------------------
4/(2+5i)
----
Multiply numerator and denominator by (2-5i) to get:
[4(2-5i)]/[4+25]
= (8-20i)/29
-----------------
i^(-35)
Applying the -1 exponent you get:
(i^3)^35
Applying the 35 exponent you get:
= i^105
= i^(104+1)
= i^(104) * i^1
= 1 * i^1
= i
=============
Cheers,
Stan H.
Answer by Nate(3500)   (Show Source): You can put this solution on YOUR website!
i^(-35)
1/i^35
1/( i * i^34 ) ~ you want an even power
1/( i * (i^2)^17 ) ~ now, you want an exponent of 2
1/( i * (-1)^17 ) ~ i^2 = -1
-1 / i ~ (-1)^17 = -1
..
4 / (2 + 5i)
4(2 - 5i) / (2 + 5i)(2 - 5i) ~ multiply the num. and dem. by the conjugate
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