Math  /  Algebra

QuestionSimplify. 8i9+8i\frac{8 i}{9+8 i}
Write your answer in the form a + bi. Reduce all fractions. \square ii 믐 Submit

Studdy Solution

STEP 1

What is this asking? We need to rewrite a fraction with a complex number in the denominator into the standard form a+bia + bi, where aa and bb are real numbers simplified as much as possible. Watch out! Remember that i2=1i^2 = -1, and don't forget to simplify the fraction at the end!

STEP 2

1. Multiply by the conjugate
2. Simplify and extract *i*
3. Separate real and imaginary parts

STEP 3

The conjugate of the denominator 9+8i9 + 8i is 98i9 - 8i.
We'll use this to get rid of the imaginary part in the denominator.

STEP 4

Multiplying the numerator and denominator by the conjugate, we get: 8i9+8i98i98i \frac{8i}{9 + 8i} \cdot \frac{9 - 8i}{9 - 8i}

STEP 5

We multiply the numerators and denominators separately: 8i(98i)9(98i)+8i(98i) \frac{8i \cdot (9 - 8i)}{9 \cdot (9 - 8i) + 8i \cdot (9 - 8i)} 72i64i28172i+72i64i2 \frac{72i - 64i^2}{81 - 72i + 72i - 64i^2}

STEP 6

Remember that i2=1i^2 = -1, so we can substitute that in: 72i64(1)8172i+72i64(1) \frac{72i - 64(-1)}{81 - 72i + 72i - 64(-1)} 72i+6481+64 \frac{72i + 64}{81 + 64}

STEP 7

Adding the real numbers in the denominator gives us: 64+72i145 \frac{64 + 72i}{145}

STEP 8

We can separate the fraction into its real and imaginary parts to get the standard form a+bia + bi: 64145+72145i \frac{64}{145} + \frac{72}{145}i

STEP 9

Our final answer in the beautiful standard form is 64145+72145i\frac{64}{145} + \frac{72}{145}i.

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