Sunday, July 14, 2013

# What are some important properties of square roots?

2:38 AM
Simplifying square roots is often required when you encounter a complicated number underneath the symbol " $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \sqrt{}$". However don’t let that scare you! When you are armed with some important properties of square roots, simplifying them is a breeze:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi {120} \sqrt{ab}=\sqrt{a} \times \sqrt{b}$
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi {120} \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
Consider the following example:

Example:
What is the value of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \sqrt{245} - \sqrt{20}$?
A) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 5\sqrt{5}$
B) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \sqrt{225}$
C) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 25\sqrt{5}$
D) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 6\sqrt{5}$
E) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 7$
Solution:

Step 1: Problems of this type rarely expect you to detemine a root of a larger number such as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 245$. In almost all cases, the expression can be simplified. In this case expressing the number inside the bracket as a multiple of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 5$ can simplify the expression:

\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \begin{align*} \sqrt{245} - \sqrt{20} &= \sqrt{ 49 \times 5} - \sqrt{4 \times 5} \\\ &= 7\sqrt{5} - 2\sqrt{5} \\\ &=(7-2)\sqrt{5} \\\ &=5\sqrt{5} \end{align*}

However be careful! You cannot apply these properties when addition or subtraction is involved. For example the following expression is correct:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue}\dpi{120}\sqrt{9+16}=\sqrt{25}=5$

However the following is clearly incorrect:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} \sqrt{9+16}=\sqrt{9}+\sqrt{16}=3+4=7$
Remember:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue}\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}$
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue}\dpi{120}\sqrt{a-b}\ne\sqrt{a}-\sqrt{b}$