If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it. - Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

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Solved Examples: Simplify Radicals

Samuel Dominic Chukwuemeka (SamDom For Peace) Calculators: Calculators for Complex Numbers
Prerequisites:
(1.) Laws of Exponents
(2.) Polynomials

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.

Simplify the radicals completely
Express your answer in terms of i

(1.) $\sqrt {-144}$


$ \sqrt {-144} \\[3ex] = \sqrt{-1} * \sqrt{144} \\[3ex] = i * 12 \\[3ex] = 12i $
(2.) $\sqrt {-21}$


$ \sqrt {-21} \\[3ex] = \sqrt{-1} * \sqrt{21} \\[3ex] = i * \sqrt{21} \\[3ex] = i\sqrt{21} $
(3.) $-\sqrt {-75}$


$ -\sqrt {-75} \\[3ex] = -1 * \sqrt{-75} \\[3ex] = - 1* \sqrt{-1} * \sqrt{75} \\[3ex] = -1 * i * \sqrt{25 * 3} \\[3ex] = -i * \sqrt{25} * \sqrt{3} \\[3ex] = -i * 5 * \sqrt{3} \\[3ex] = -5i\sqrt{3} $
(4.) $-\sqrt {-53}$


$ -\sqrt {-53} \\[3ex] = -1 * \sqrt{-1} * \sqrt{53} \\[3ex] = - 1* i * \sqrt{53} \\[3ex] = -i\sqrt{53} $
(5.) $\sqrt {-150}$


$ \sqrt {-150} \\[3ex] = \sqrt{-1} * \sqrt{150} \\[3ex] = i * \sqrt{25 * 6} \\[3ex] = i * \sqrt{25} * \sqrt{6} \\[3ex] = i * 5 * \sqrt{6} \\[3ex] = 5i\sqrt{6} $
(6.) $\sqrt{\dfrac{-49}{75}}$


$ \sqrt{\dfrac{-49}{75}} \\[5ex] = \sqrt {\dfrac{-1 * 49}{75}} \\[5ex] = \sqrt{-1} * \sqrt{\dfrac{49}{75}} \\[5ex] = i * \dfrac{\sqrt{49}}{\sqrt{75}} \\[5ex] = i * \dfrac{7}{\sqrt{25 * 3}} \\[5ex] = i * \dfrac{7}{\sqrt{25} * \sqrt{3}} \\[5ex] = i * \dfrac{7}{5 * \sqrt{3}} \\[5ex] = \dfrac{7i}{5\sqrt{3}} \\[5ex] = \dfrac{7i}{5\sqrt{3}} * \dfrac{\sqrt{3}}{\sqrt{3}} \\[5ex] = \dfrac{7i\sqrt{3}}{5\sqrt{3}*\sqrt{3}} \\[5ex] = \dfrac{7i\sqrt{3}}{5(3)} \\[5ex] = \dfrac{7i\sqrt{3}}{15} \\[5ex] $
(7.) $(\sqrt {-36})(\sqrt{-4})$


$ (\sqrt {-36})(\sqrt{-4}) \\[3ex] = (\sqrt{-1 * 36})(\sqrt{-1 * 4}) \\[3ex] = \sqrt{-1} * \sqrt{36} * \sqrt{-1} * \sqrt{4} \\[3ex] = i * 6 * i * 2 \\[3ex] = i^2 * 12 \\[3ex] = -1 * 12 \\[3ex] = -12 $
(8.)


$ -2\left|3 - \dfrac{p}{3}\right| + 1 = -5 \\[5ex] -2\left|3 - \dfrac{p}{3}\right| = -5 - 1 \\[5ex] -2\left|3 - \dfrac{p}{3}\right| = -6 \\[5ex] \left|3 - \dfrac{p}{3}\right| = \dfrac{-6}{-2} \\[5ex] \left|3 - \dfrac{p}{3}\right| = 3 \\[5ex] This\:\: means\:\: that \\[5ex] 3 - \dfrac{p}{3} = 3 \:\:OR\:\: -(3 - \dfrac{p}{3} = 3 \\[5ex] 3 - \dfrac{p}{3} = 3 \\[5ex] 3 - 3 = \dfrac{p}{3} \\[5ex] 0 = \dfrac{p}{3} \\[5ex] \dfrac{p}{3} = 0 \\[5ex] p = 0(3) \\[3ex] p = 0 \\[3ex] OR \\[3ex] -(3 - \dfrac{p}{3} = 3 \\[5ex] Divide\:\: both\:\: sides\:\: by\:\: -1 \\[3ex] 3 - \dfrac{p}{3} = -3 \\[5ex] 3 + 3 = \dfrac{p}{3} \\[5ex] 6 = \dfrac{p}{3} \\[5ex] \dfrac{p}{3} = 6 \\[5ex] p = 6(3) \\[3ex] p = 18 \\[3ex] $ Check
Check for both values.
$ \underline{LHS} \\[3ex] -2\left|3 - \dfrac{p}{3}\right| + 1 \\[5ex] p = 0 \\[3ex] -2\left|3 - \dfrac{0}{3}\right| + 1 \\[5ex] -2|3 - 0| + 1 \\[5ex] -2|3| + 1 \\[3ex] -2(3) + 1 \\[3ex] -6 + 1 \\[3ex] -5 \\[3ex] $ $p = 0$ is a solution



$ -2\left|3 - \dfrac{p}{3}\right| + 1 \\[5ex] p = 18 \\[3ex] -2\left|3 - \dfrac{18}{3}\right| + 1 \\[5ex] -2|3 - 6| + 1 \\[5ex] -2|-3| + 1 \\[3ex] -2(3) + 1 \\[3ex] -6 + 1 \\[3ex] -5 \\[3ex] $ $p = 18$ is a solution 		$ \underline{RHS} \\[3ex] -5 $