Simplifying Expressions - GMAT Math Study Guide
Table of Contents
Division
In working with division problems, factoring can greatly simplify an expression as common terms can be canceled out, making for significantly easier and quicker computation.
In working problems without a calculator, spotting and removing common terms greatly increases the speed at which you can find an answer. Consider the following example:
Examples
7 is a common factor that can be removed to simplify the following equation.
Another example:
Another example:
Multiplication
In dealing with multiplication problems, it is often possible to simplify an expression by finding a common factor and dividing the entire equation by it.
Examples
4(2x + 25y + 6z + 11) = 4(100)
2x + 25y + 6z + 11 = 100 [Divided both sides by 4]
Another example:
3(9x) + 3(5y) = 3(10)
3(9x + 5y) = 3(10)
9x + 5y = 10 [Divided both sides by 3]
Addition & Subtraction
In approaching problems that involve addition and subtraction, it is important to look for like terms that can be simplified through consolidation.
10x + 5y + 20z + 2x + 6y + 9z - 2x - 3y - 4z + 15x + 7y + 20z = 0
10x + 2x - 32 + 15x + 5y + 6y - 3y + 7y + 20z + 9z - 4z + 20z = 0
(10x + 2x - 2x + 15x) + (5y + 6y - 3y + 7y) + (20z + 9z - 4z + 20z) = 0
25x + 15y + 45z = 0
5(5x + 3y + 9z) = 0
5x + 3y + 9z = 0
Another example:
75x2 + 75x + 150 = 0
x2 + x + 2 = 0 [Using Division]
Radicals
In working with radicals, there are a number of different techniques. Unfortunately, there are no rules and some problems take a guess-and-check approach to solve. Nonetheless, the following strategies for factoring radicals help: (a) look for all the perfect squares (b) combine like terms (c) factor out common terms.
In working the following example, the key to unlocking the problem is noticing that common factors of 7 and 5 are at work.
Types of GMAT Problems
- Solve by Factoring and Simplifying
Recall that any number or expression divided by itself is 1. There is, however, one exception. If the expression can possibly be zero, it can only be written as 1 if a statement is included which excludes all possible values which would cause the expression to be zero.
Correct Answer: E- The numerator, which is a difference of squares, can be factored.
x2-4 = (x+2)*(x-2). - Because there is a x+2 in both the numerator and denominator, it can be canceled and rewritten as 1.
- Thus, the equation can be rewritten as x-2 = 0.
- By adding 2 to both sides of the equation, we see that x=2.
- The numerator, which is a difference of squares, can be factored.
- Simplify Square Roots
To simplify a square root, factor out all of the perfect squares. Then take the square root of all the perfect squares and place them in front of the square root. If there were multiple perfect squares, multiply their square roots together.
Correct Answer: D- Factor the perfect squares out of 900.
900 = 302
Or, if smaller squares are used
900 = 100*9 = 102*32 - Take the root of each of the perfect squares.
- Factor the perfect squares out of 900.