Studdy AI Logo
Math  /  Algebra

QuestionSimplify. 2+10i5+8i\frac{2+10 i}{5+8 i}
Write your answer in the form a + bi. Reduce all fractions. □\square

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with complex numbers and write the answer in the standard form a+bia + bi, where aa and bb are real numbers. Watch out! Remember that i2=−1i^2 = -1.
Don't forget to reduce the fractions at the end!

STEP 2

1. Multiply by the conjugate
2. Simplify and separate

STEP 3

To get rid of the imaginary part in the denominator, we'll **multiply** both the numerator and denominator by the **conjugate** of the denominator.
The conjugate of 5+8i5 + 8i is 5−8i5 - 8i.
Remember, we change the sign of the imaginary part!

STEP 4

2+10i5+8i⋅5−8i5−8i \frac{2 + 10i}{5 + 8i} \cdot \frac{5 - 8i}{5 - 8i} Multiplying by 5−8i5−8i\frac{5 - 8i}{5 - 8i} is like multiplying by **one**, so it doesn't change the *value* of our expression, just how it looks!

STEP 5

(2+10i)(5−8i)=2⋅5+2⋅(−8i)+10i⋅5+10i⋅(−8i)=10−16i+50i−80i2 (2 + 10i)(5 - 8i) = 2 \cdot 5 + 2 \cdot (-8i) + 10i \cdot 5 + 10i \cdot (-8i) = 10 - 16i + 50i - 80i^2 Since i2=−1i^2 = -1, we have 10−16i+50i−80i2=10−16i+50i−80(−1)=10−16i+50i+80=90+34i 10 - 16i + 50i - 80i^2 = 10 - 16i + 50i - 80(-1) = 10 - 16i + 50i + 80 = 90 + 34i

STEP 6

(5+8i)(5−8i)=5⋅5+5⋅(−8i)+8i⋅5+8i⋅(−8i)=25−40i+40i−64i2 (5 + 8i)(5 - 8i) = 5 \cdot 5 + 5 \cdot (-8i) + 8i \cdot 5 + 8i \cdot (-8i) = 25 - 40i + 40i - 64i^2 The middle terms add to zero, which is exactly what we want! 25−64i2=25−64(−1)=25+64=89 25 - 64i^2 = 25 - 64(-1) = 25 + 64 = 89

STEP 7

So, our fraction becomes: 90+34i89 \frac{90 + 34i}{89}

STEP 8

We want our answer in the form a+bia + bi, so let's separate the fraction: 90+34i89=9089+3489i \frac{90 + 34i}{89} = \frac{90}{89} + \frac{34}{89}i

STEP 9

Can we simplify the fractions 9089\frac{90}{89} and 3489\frac{34}{89}?
Nope! 89 is a prime number, and neither 90 nor 34 is divisible by 89.
So, we're done!

STEP 10

Our final answer is 9089+3489i\frac{90}{89} + \frac{34}{89}i.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord