Question

explain simplifying square roots with negative numbers
read explain parts a and b and complete your turn #1 - 2 below (adapted from lesson 3.1).
simplifying square roots with negative numbers links your skills in quadratic equations and imaginary numbers.
this skill deepens your understanding of complex numbers, strengthening your ability to handle advanced
algebraic manipulations effectively. use imaginary numbers to simplify square roots with negative numbers.
imaginary numbers are the square roots of negative numbers. this means $sqrt{-1}=i$. the imaginary unit can be
written in the form $bi$ where $b$ is a nonzero real number and $i$ is the imaginary unit that represents $sqrt{-1}$.
example
simplify
a. $sqrt{-36}$
$=sqrt{-1cdot36}$
$=sqrt{-1}cdotsqrt{36}$
$=isqrt{36}$
$=i6$
$6i$
simplify the radical by seeing - 36 as factors - 1 and 36
substitute $sqrt{-1}$ for $i$
$sqrt{36}$ equals 6
write the imaginary unit in the form $bi$
b. $sqrt{-75}$
$=sqrt{-1cdot75}$
$=sqrt{-1}cdotsqrt{75}$
$=isqrt{75}$
$=isqrt{3cdot25}$
$=isqrt{3}cdotsqrt{25}$
$=i5sqrt{3}$
$5isqrt{3}$
simplify the radical by seeing - 75 as factors - 1 and 75
substitute $sqrt{-1}$ for $i$
simplify the radical by seeing 75 as factors 3 and 25
$sqrt{3}$ remains under the square root and $sqrt{25}$ equals 5
write the imaginary unit in the form $bi$
your turn
simplify

1. $sqrt{-18}$
2. $sqrt{-200}$
3. $sqrt{-64}$

Explanation:

Step1: Factor -18

$\sqrt{-18}=\sqrt{-1\times18}$

Step2: Split the square - root

$=\sqrt{-1}\times\sqrt{18}$

Step3: Substitute $\sqrt{-1}$ with $i$

$=i\sqrt{18}$

Step4: Simplify $\sqrt{18}$

$=i\sqrt{9\times2}=i\sqrt{9}\times\sqrt{2}$

Step5: Calculate $\sqrt{9}$

$=i\times3\sqrt{2}=3i\sqrt{2}$

Step1: Factor -200

$\sqrt{-200}=\sqrt{-1\times200}$

Step2: Split the square - root

$=\sqrt{-1}\times\sqrt{200}$

Step3: Substitute $\sqrt{-1}$ with $i$

$=i\sqrt{200}$

Step4: Simplify $\sqrt{200}$

$=i\sqrt{100\times2}=i\sqrt{100}\times\sqrt{2}$

Step5: Calculate $\sqrt{100}$

$=i\times10\sqrt{2}=10i\sqrt{2}$

Step1: Factor -64

$\sqrt{-64}=\sqrt{-1\times64}$

Step2: Split the square - root

$=\sqrt{-1}\times\sqrt{64}$

Step3: Substitute $\sqrt{-1}$ with $i$

$=i\sqrt{64}$

Step4: Calculate $\sqrt{64}$

$=i\times8 = 8i$

Answer:

$3i\sqrt{2}$