Math / Algebra

QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $-18+\sqrt{-5}$ $\square$ i $\sqrt{ }$ Submit

Studdy Solution

STEP 1

What is this asking? We need to rewrite $-18 + \sqrt{-5}$ in the standard complex number form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit. Watch out! Remember that the square root of a negative number involves the imaginary unit $i$ , where $i = \sqrt{-1}$ .
Don't forget to simplify any radicals if possible!

STEP 2

1. Rewrite the square root of a negative number.
2. Simplify and write in standard form.

STEP 3

We're given the expression $-18 + \sqrt{-5}$ .
Our first step is to deal with that pesky square root of a negative number.
Remember, we can rewrite $\sqrt{-n}$ as $\sqrt{-1} \cdot \sqrt{n}$ .
Since $i = \sqrt{-1}$ , we can then write it as $i\sqrt{n}$ .

STEP 4

So, let's apply this to our problem.
We have $\sqrt{-5}$ .
We can rewrite this as $\sqrt{-1 \cdot 5}$ , which simplifies to $\sqrt{-1} \cdot \sqrt{5}$ .
Since $\sqrt{-1}$ is the same as $i$ , we get $i\sqrt{5}$ !

STEP 5

Now, we can substitute this back into our original expression: $-18 + \sqrt{-5}$ becomes $-18 + i\sqrt{5}$ .

STEP 6

This is almost in standard complex number form, $a + bi$ .
Here, our real part, $a$ , is -18, and our imaginary part, $b$ , is $\sqrt{5}$ .
So, our final answer is $-18 + i\sqrt{5}$ .
We can't simplify the radical any further, so we're done!

STEP 7

$-18 + i\sqrt{5}$

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QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $-18+\sqrt{-5}$ $\square$ i $\sqrt{ }$ Submit

Studdy Solution

STEP 1

STEP 2

1. Rewrite the square root of a negative number.
2. Simplify and write in standard form.

STEP 3

We're given the expression $-18 + \sqrt{-5}$ .
Our first step is to deal with that pesky square root of a negative number.
Remember, we can rewrite $\sqrt{-n}$ as $\sqrt{-1} \cdot \sqrt{n}$ .
Since $i = \sqrt{-1}$ , we can then write it as $i\sqrt{n}$ .

STEP 4

So, let's apply this to our problem.
We have $\sqrt{-5}$ .
We can rewrite this as $\sqrt{-1 \cdot 5}$ , which simplifies to $\sqrt{-1} \cdot \sqrt{5}$ .
Since $\sqrt{-1}$ is the same as $i$ , we get $i\sqrt{5}$ !

STEP 5

Now, we can substitute this back into our original expression: $-18 + \sqrt{-5}$ becomes $-18 + i\sqrt{5}$ .

STEP 6

This is almost in standard complex number form, $a + bi$ .
Here, our real part, $a$ , is -18, and our imaginary part, $b$ , is $\sqrt{5}$ .
So, our final answer is $-18 + i\sqrt{5}$ .
We can't simplify the radical any further, so we're done!

STEP 7

$-18 + i\sqrt{5}$

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QuestionUse the imaginary number iii to rewrite the expression below as a complex number. Simplify all radicals. −18+−5-18+\sqrt{-5}−18+−5​ □\square□ i \sqrt{ }​ Submit

Studdy Solution

STEP 1

STEP 2

1. Rewrite the square root of a negative number.2. Simplify and write in standard form.

STEP 3

STEP 4

So, let's apply this to our problem.We have −5\sqrt{-5}−5​.We can rewrite this as −1⋅5\sqrt{-1 \cdot 5}−1⋅5​, which simplifies to −1⋅5\sqrt{-1} \cdot \sqrt{5}−1​⋅5​.Since −1\sqrt{-1}−1​ is the same as iii, we get i5i\sqrt{5}i5​!

STEP 5

Now, we can substitute this back into our original expression: −18+−5-18 + \sqrt{-5}−18+−5​ becomes −18+i5-18 + i\sqrt{5}−18+i5​.

STEP 6

This is almost in standard complex number form, a+bia + bia+bi.Here, our **real part**, aaa, is **-18**, and our **imaginary part**, bbb, is 5\sqrt{5}5​.So, our **final answer** is −18+i5-18 + i\sqrt{5}−18+i5​.We can't simplify the radical any further, so we're done!

STEP 7

−18+i5-18 + i\sqrt{5}−18+i5​

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QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $-18+\sqrt{-5}$ $\square$ i $\sqrt{ }$ Submit

1. Rewrite the square root of a negative number.
2. Simplify and write in standard form.

So, let's apply this to our problem.
We have $\sqrt{-5}$ .
We can rewrite this as $\sqrt{-1 \cdot 5}$ , which simplifies to $\sqrt{-1} \cdot \sqrt{5}$ .
Since $\sqrt{-1}$ is the same as $i$ , we get $i\sqrt{5}$ !

Now, we can substitute this back into our original expression: $-18 + \sqrt{-5}$ becomes $-18 + i\sqrt{5}$ .

This is almost in standard complex number form, $a + bi$ .
Here, our real part, $a$ , is -18, and our imaginary part, $b$ , is $\sqrt{5}$ .
So, our final answer is $-18 + i\sqrt{5}$ .
We can't simplify the radical any further, so we're done!

$-18 + i\sqrt{5}$