Math / Algebra

QuestionSimplify. $-\left(\frac{8 i}{9}-\frac{8}{3}\right)-\left(3 i-\frac{8}{9}\right)+\frac{2+i}{3}$
Write your answer in the form a + bi. Simplify all fractions. $\square$

Studdy Solution

STEP 1

What is this asking? We need to simplify a complex expression with fractions and imaginary numbers, and write the final answer in the standard a + bi form. Watch out! Don't forget to distribute the negative signs correctly and be careful when adding and subtracting fractions – make sure those denominators match up!

STEP 2

1. Distribute the negative signs
2. Group like terms
3. Simplify real and imaginary parts
4. Combine and write in standard form

STEP 3

Alright, let's kick things off by distributing those negative signs!
We have $-\left(\frac{8i}{9} - \frac{8}{3}\right) - \left(3i - \frac{8}{9}\right) + \frac{2+i}{3}$ .
Distributing the negative signs to the terms inside the parentheses gives us $-\frac{8i}{9} + \frac{8}{3} - 3i + \frac{8}{9} + \frac{2+i}{3}$ .
Remember, multiplying by -1 flips the sign of each term inside the parentheses.

STEP 4

Now, let's get organized!
We'll group the real parts and the imaginary parts separately.
Our expression becomes $\left(\frac{8}{3} + \frac{8}{9} + \frac{2}{3}\right) + \left(-\frac{8i}{9} - 3i + \frac{i}{3}\right)$ .
See how we've neatly separated the terms with i from the terms without i?

STEP 5

Time to tackle those fractions!
For the real part, we need a common denominator, which is 9.
So, we rewrite $\frac{8}{3}$ as $\frac{8 \cdot 3}{3 \cdot 3} = \frac{24}{9}$ and $\frac{2}{3}$ as $\frac{2 \cdot 3}{3 \cdot 3} = \frac{6}{9}$ .
Now, the real part is $\frac{24}{9} + \frac{8}{9} + \frac{6}{9} = \frac{24+8+6}{9} = \frac{38}{9}$ .

STEP 6

Onto the imaginary part!
Again, we need a common denominator of 9.
We rewrite $3i$ as $\frac{3i \cdot 9}{1 \cdot 9} = \frac{27i}{9}$ and $\frac{i}{3}$ as $\frac{i \cdot 3}{3 \cdot 3} = \frac{3i}{9}$ .
Now, the imaginary part is $-\frac{8i}{9} - \frac{27i}{9} + \frac{3i}{9} = \frac{-8i - 27i + 3i}{9} = \frac{(-8-27+3)i}{9} = \frac{-32i}{9}$ .

STEP 7

Final stretch!
Combining the simplified real and imaginary parts gives us $\frac{38}{9} - \frac{32i}{9}$ .
This is exactly the a + bi form we were aiming for, where $a = \frac{38}{9}$ and $b = -\frac{32}{9}$ .

STEP 8

Our simplified expression is $\frac{38}{9} - \frac{32}{9}i$ .

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QuestionSimplify. $-\left(\frac{8 i}{9}-\frac{8}{3}\right)-\left(3 i-\frac{8}{9}\right)+\frac{2+i}{3}$
Write your answer in the form a + bi. Simplify all fractions. $\square$

Studdy Solution

STEP 1

STEP 2

1. Distribute the negative signs
2. Group like terms
3. Simplify real and imaginary parts
4. Combine and write in standard form

STEP 3

STEP 4

Now, let's get organized!
We'll group the real parts and the imaginary parts separately.
Our expression becomes $\left(\frac{8}{3} + \frac{8}{9} + \frac{2}{3}\right) + \left(-\frac{8i}{9} - 3i + \frac{i}{3}\right)$ .
See how we've neatly separated the terms with i from the terms without i?

STEP 5

STEP 6

STEP 7

Final stretch!
Combining the simplified real and imaginary parts gives us $\frac{38}{9} - \frac{32i}{9}$ .
This is exactly the a + bi form we were aiming for, where $a = \frac{38}{9}$ and $b = -\frac{32}{9}$ .

STEP 8

Our simplified expression is $\frac{38}{9} - \frac{32}{9}i$ .

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QuestionSimplify. −(8i9−83)−(3i−89)+2+i3-\left(\frac{8 i}{9}-\frac{8}{3}\right)-\left(3 i-\frac{8}{9}\right)+\frac{2+i}{3}−(98i​−38​)−(3i−98​)+32+i​Write your answer in the form a + bi. Simplify all fractions. □\square□

Studdy Solution

STEP 1

STEP 2

1. Distribute the negative signs2. Group like terms3. Simplify real and imaginary parts4. Combine and write in standard form

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

**Final stretch**!Combining the simplified real and imaginary parts gives us 389−32i9\frac{38}{9} - \frac{32i}{9}938​−932i​.This is exactly the *a* + *bi* form we were aiming for, where a=389a = \frac{38}{9}a=938​ and b=−329b = -\frac{32}{9}b=−932​.

STEP 8

Our simplified expression is 389−329i\frac{38}{9} - \frac{32}{9}i938​−932​i.

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Start learning now

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QuestionSimplify. $-\left(\frac{8 i}{9}-\frac{8}{3}\right)-\left(3 i-\frac{8}{9}\right)+\frac{2+i}{3}$
Write your answer in the form a + bi. Simplify all fractions. $\square$

1. Distribute the negative signs
2. Group like terms
3. Simplify real and imaginary parts
4. Combine and write in standard form

Final stretch!
Combining the simplified real and imaginary parts gives us $\frac{38}{9} - \frac{32i}{9}$ .
This is exactly the a + bi form we were aiming for, where $a = \frac{38}{9}$ and $b = -\frac{32}{9}$ .

Our simplified expression is $\frac{38}{9} - \frac{32}{9}i$ .