Math / Algebra

QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-77}$ $\square$ i $\square$ Submit

Studdy Solution

STEP 1

What is this asking? We need to rewrite the square root of a negative number using the imaginary unit $i$ . Watch out! Remember that $\sqrt{-1}$ is defined as $i$ , and be careful when simplifying radicals!

STEP 2

1. Rewrite the expression
2. Simplify the radical
3. Write as a complex number

STEP 3

Let's rewrite the expression $\sqrt{-77}$ using the definition of the imaginary unit $i$ .
Remember that $i = \sqrt{-1}$ , so we can rewrite $\sqrt{-77}$ as: $\sqrt{-77} = \sqrt{-1 \cdot 77}$

STEP 4

Now, we can use the product property of square roots, which says $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ .
So, we get: $\sqrt{-1 \cdot 77} = \sqrt{-1} \cdot \sqrt{77}$

STEP 5

Since $\sqrt{-1}$ is defined as $i$ , we can substitute $i$ for $\sqrt{-1}$ : $\sqrt{-1} \cdot \sqrt{77} = i \cdot \sqrt{77}$

STEP 6

Now, let's see if we can simplify $\sqrt{77}$ .
We look for perfect square factors of $77$ .
We can write $77$ as $7 \cdot 11$ .
Since $7$ and $11$ are both prime numbers, there are no perfect square factors other than $1$ .
So, $\sqrt{77}$ is already in its simplest radical form!

STEP 7

We've got $i \cdot \sqrt{77}$ , which is usually written as $\sqrt{77}i$ .
This is a complex number in the form $a + bi$ , where $a = 0$ is the real part and $\sqrt{77}$ is the imaginary part.
So, our final answer is $\sqrt{77}i$ .

STEP 8

$\sqrt{77}$ i

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QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-77}$ $\square$ i $\square$ Submit

Studdy Solution

STEP 1

What is this asking? We need to rewrite the square root of a negative number using the imaginary unit $i$ . Watch out! Remember that $\sqrt{-1}$ is defined as $i$ , and be careful when simplifying radicals!

STEP 2

1. Rewrite the expression
2. Simplify the radical
3. Write as a complex number

STEP 3

Let's rewrite the expression $\sqrt{-77}$ using the definition of the imaginary unit $i$ .
Remember that $i = \sqrt{-1}$ , so we can rewrite $\sqrt{-77}$ as: $\sqrt{-77} = \sqrt{-1 \cdot 77}$

STEP 4

Now, we can use the product property of square roots, which says $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ .
So, we get: $\sqrt{-1 \cdot 77} = \sqrt{-1} \cdot \sqrt{77}$

STEP 5

Since $\sqrt{-1}$ is defined as $i$ , we can substitute $i$ for $\sqrt{-1}$ : $\sqrt{-1} \cdot \sqrt{77} = i \cdot \sqrt{77}$

STEP 6

Now, let's see if we can simplify $\sqrt{77}$ .
We look for perfect square factors of $77$ .
We can write $77$ as $7 \cdot 11$ .
Since $7$ and $11$ are both prime numbers, there are no perfect square factors other than $1$ .
So, $\sqrt{77}$ is already in its simplest radical form!

STEP 7

We've got $i \cdot \sqrt{77}$ , which is usually written as $\sqrt{77}i$ .
This is a complex number in the form $a + bi$ , where $a = 0$ is the real part and $\sqrt{77}$ is the imaginary part.
So, our final answer is $\sqrt{77}i$ .

STEP 8

$\sqrt{77}$ i

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QuestionUse the imaginary number iii to rewrite the expression below as a complex number. Simplify all radicals. −77\sqrt{-77}−77​ □\square□ i □\square□ Submit

Studdy Solution

STEP 1

What is this asking? We need to rewrite the square root of a negative number using the imaginary unit iii. Watch out! Remember that −1\sqrt{-1}−1​ is defined as iii, and be careful when simplifying radicals!

STEP 2

1. Rewrite the expression2. Simplify the radical3. Write as a complex number

STEP 3

Let's **rewrite** the expression −77\sqrt{-77}−77​ using the **definition** of the imaginary unit iii.Remember that i=−1i = \sqrt{-1}i=−1​, so we can rewrite −77\sqrt{-77}−77​ as: −77=−1⋅77 \sqrt{-77} = \sqrt{-1 \cdot 77} −77​=−1⋅77​

STEP 4

Now, we can use the **product property of square roots**, which says a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}a⋅b​=a​⋅b​.So, we get: −1⋅77=−1⋅77 \sqrt{-1 \cdot 77} = \sqrt{-1} \cdot \sqrt{77} −1⋅77​=−1​⋅77​

STEP 5

Since −1\sqrt{-1}−1​ is defined as iii, we can **substitute** iii for −1\sqrt{-1}−1​: −1⋅77=i⋅77 \sqrt{-1} \cdot \sqrt{77} = i \cdot \sqrt{77} −1​⋅77​=i⋅77​

STEP 6

STEP 7

We've got i⋅77i \cdot \sqrt{77}i⋅77​, which is usually written as 77i\sqrt{77}i77​i.This is a **complex number** in the form a+bia + bia+bi, where a=0a = 0a=0 is the **real part** and 77\sqrt{77}77​ is the **imaginary part**.So, our final answer is 77i\sqrt{77}i77​i.

STEP 8

77\sqrt{77}77​ i

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QuestionUse the imaginary number $i$ to rewrite the expression below as a complex number. Simplify all radicals. $\sqrt{-77}$ $\square$ i $\square$ Submit

What is this asking? We need to rewrite the square root of a negative number using the imaginary unit $i$ . Watch out! Remember that $\sqrt{-1}$ is defined as $i$ , and be careful when simplifying radicals!

1. Rewrite the expression
2. Simplify the radical
3. Write as a complex number

Let's rewrite the expression $\sqrt{-77}$ using the definition of the imaginary unit $i$ .
Remember that $i = \sqrt{-1}$ , so we can rewrite $\sqrt{-77}$ as: $\sqrt{-77} = \sqrt{-1 \cdot 77}$

Now, we can use the product property of square roots, which says $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ .
So, we get: $\sqrt{-1 \cdot 77} = \sqrt{-1} \cdot \sqrt{77}$

Since $\sqrt{-1}$ is defined as $i$ , we can substitute $i$ for $\sqrt{-1}$ : $\sqrt{-1} \cdot \sqrt{77} = i \cdot \sqrt{77}$

We've got $i \cdot \sqrt{77}$ , which is usually written as $\sqrt{77}i$ .
This is a complex number in the form $a + bi$ , where $a = 0$ is the real part and $\sqrt{77}$ is the imaginary part.
So, our final answer is $\sqrt{77}i$ .

$\sqrt{77}$ i