Sunday, July 14, 2013

# What are the 5 exponent rules on GRE that everyone must remember?

10:47 AM
In order to solve any question involving exponents you must remember the 5 basic rules of exponents. Remember them? If not we will help you jog your memory:

 Exponent Multiplication Rule: $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x^m \times x^n=x^{m+n}$ Exponent Division Rule: $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x^m \div x^n = x^{m-n}$ Raising power to power Rule: $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} (x^m)^n=x^{m \times n}$ Common Exponent Multiplication $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x^m \times y ^m = (xy)^m$ Common Exponent Division $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \frac{x^m}{y^m}=( \frac{x}{y} \left) ^m$
Exponent rules are useful for solving any test problem that involves power. Here is a quick example:
Example:
What is the value of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$ if  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 4^{5x-2}=16^{x+2}$?
A) 1
B) 2
C) 3
D) 4
E) 5
Solution:

Step 1: The bases on both sides are not equal. However we know that $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 16=4^2$,
Therefore $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 16^{x+2} = (4^2)^{x+2} = 4^{2(x+2)}$
Step 2: Our equation can now be written as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 4^{5x-2}=4^{2(x+2)}$. Since the bases are equal, we can equate the exponents and solve the linear equation:
\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi {120} \begin{align*} 5x-2&=2(x+2) \\ 5x-2&=2x+4 \\ 5x-2x &=4+2 \\ 3x&=6 \\ \Rightarrow x&=2 \end{align*}
B is the correct answer.
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