Math / Algebra

QuestionAdd and simplify. $\begin{array}{l} (4+\sqrt{-16})+(9+\sqrt{-36}) \\ (4+\sqrt{-16})+(9+\sqrt{-36})= \end{array}$ $\square$ (Simplify your answer. Type your answer in the form a +bi .)

Studdy Solution

STEP 1

What is this asking? We're adding two complex numbers and simplifying the result. Watch out! Remember that the square root of a negative number involves the imaginary unit $i$ .

STEP 2

1. Simplify the square roots
2. Group like terms
3. Combine like terms

STEP 3

Let's rewrite the square roots of negative numbers in terms of $i$ .
Remember that $i = \sqrt{-1}$ , so we can rewrite $\sqrt{-16}$ as $\sqrt{16} \cdot \sqrt{-1} = 4i$ .
Similarly, we can rewrite $\sqrt{-36}$ as $\sqrt{36} \cdot \sqrt{-1} = 6i$ .
So, we have $(4 + 4i) + (9 + 6i)$ .
This makes things much clearer!

STEP 4

Now, let's group the real parts and the imaginary parts together.
We have the real parts $4$ and $9$ , and the imaginary parts $4i$ and $6i$ .
We can rewrite our expression as $(4 + 9) + (4i + 6i)$ .
This sets us up perfectly for the next step!

STEP 5

Finally, let's combine those like terms.
Adding the real parts, $4 + 9 = 13$ .
Adding the imaginary parts, $4i + 6i = 10i$ .
So, our simplified answer is $13 + 10i$ .
Awesome!

STEP 6

$13 + 10i$

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QuestionAdd and simplify. $\begin{array}{l} (4+\sqrt{-16})+(9+\sqrt{-36}) \\ (4+\sqrt{-16})+(9+\sqrt{-36})= \end{array}$ $\square$ (Simplify your answer. Type your answer in the form a +bi .)

Studdy Solution

STEP 1

What is this asking? We're adding two complex numbers and simplifying the result. Watch out! Remember that the square root of a negative number involves the imaginary unit $i$ .

STEP 2

1. Simplify the square roots
2. Group like terms
3. Combine like terms

STEP 3

STEP 4

Now, let's group the real parts and the imaginary parts together.
We have the real parts $4$ and $9$ , and the imaginary parts $4i$ and $6i$ .
We can rewrite our expression as $(4 + 9) + (4i + 6i)$ .
This sets us up perfectly for the next step!

STEP 5

Finally, let's combine those like terms.
Adding the real parts, $4 + 9 = 13$ .
Adding the imaginary parts, $4i + 6i = 10i$ .
So, our simplified answer is $13 + 10i$ .
Awesome!

STEP 6

$13 + 10i$

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QuestionAdd and simplify. (4+−16)+(9+−36)(4+−16)+(9+−36)=\begin{array}{l} (4+\sqrt{-16})+(9+\sqrt{-36}) \\ (4+\sqrt{-16})+(9+\sqrt{-36})= \end{array}(4+−16​)+(9+−36​)(4+−16​)+(9+−36​)=​ □\square□ (Simplify your answer. Type your answer in the form a +bi .)

Studdy Solution

STEP 1

What is this asking? We're adding two complex numbers and simplifying the result. Watch out! Remember that the square root of a negative number involves the imaginary unit iii.

STEP 2

1. Simplify the square roots2. Group like terms3. Combine like terms

STEP 3

STEP 4

Now, let's **group** the real parts and the imaginary parts together.We have the real parts 444 and 999, and the imaginary parts 4i4i4i and 6i6i6i.We can rewrite our expression as (4+9)+(4i+6i)(4 + 9) + (4i + 6i)(4+9)+(4i+6i).This sets us up perfectly for the next step!

STEP 5

Finally, let's **combine** those like terms.Adding the real parts, 4+9=134 + 9 = 134+9=13.Adding the imaginary parts, 4i+6i=10i4i + 6i = 10i4i+6i=10i.So, our simplified answer is 13+10i13 + 10i13+10i.Awesome!

STEP 6

13+10i13 + 10i13+10i

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QuestionAdd and simplify. $\begin{array}{l} (4+\sqrt{-16})+(9+\sqrt{-36}) \\ (4+\sqrt{-16})+(9+\sqrt{-36})= \end{array}$ $\square$ (Simplify your answer. Type your answer in the form a +bi .)

What is this asking? We're adding two complex numbers and simplifying the result. Watch out! Remember that the square root of a negative number involves the imaginary unit $i$ .

1. Simplify the square roots
2. Group like terms
3. Combine like terms

Now, let's group the real parts and the imaginary parts together.
We have the real parts $4$ and $9$ , and the imaginary parts $4i$ and $6i$ .
We can rewrite our expression as $(4 + 9) + (4i + 6i)$ .
This sets us up perfectly for the next step!

Finally, let's combine those like terms.
Adding the real parts, $4 + 9 = 13$ .
Adding the imaginary parts, $4i + 6i = 10i$ .
So, our simplified answer is $13 + 10i$ .
Awesome!

$13 + 10i$