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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Simplifying Fractions

Examples with solutions

 

Example 1:

Simplify .

solution:

Since the numerator is the sum of two terms, we must begin by factoring it as much as possible:

(both terms in the original expression share a factor of 3)

(both terms in the square brackets of the preceding line share a factor of x )

(both terms in the square brackets of the preceding line share a factor of y )

Thus

as the final answer.

The initial factoring step is absolutely essential. If instead you decided to begin by cancelling between the denominator and the first term of the numerator, along the lines of

you would end up with a completely wrong result. This kind of thing is not wrong so much because we say so, or because you didn’t use “our” method, the method “we” told you to use. Rather, it is wrong because the result does not have the same mathematical meaning, nor will it give the same mathematical results as the original expression, and so it’s about the same sort of thing as deciding the change the meaning of words when you speak (but pretending you haven’t!). If you substitute specific pair of values for x and y into this proposed simplified expression, you will get different value from it than you would get when you substitute the same values of x and y into the original expression. Simplification is intended to give a mathematical expression which is easier to write down and work with than the original expression, but which gives exactly the same results as the original expression in all circumstances. If our apparent simplified expression does not do this, then we have made a mistake.

 

Example 2:

Simplify .

solution:

It is so tempting just to cancel the ‘5x 2’ terms and be done:

or even

By now you’re screaming back, “No! No! Both of these results are wrong! Wrong! Wrong!” In both cases, the that is being cancelled is not a factor in the denominator (it is a “term” in the denominator) and hence cannot be validly cancelled. In the second version above, another error is made as well. When factors are cancelled, they always leave a result of “1” behind. If it is a multiplication by 1, we need not write down the 1, as in

But, if it is an “added” 1, you cannot drop it:

(1)(20x) = 20x but (1) + (20x) 20x

This latter is as incorrect as writing

This is supposed to look like a fraction with nothing written in the numerator or in the denominator (that is, the ultimate in cancellation – perhaps we should consider this to be annihilation!). Of course, you can easily see this is nonsense. Anyway, back to the original example. Before any cancelling can be contemplated, both the numerator and denominator must be factored completely. The numerator is already in factored form. However, for the denominator we have

5x 2 + 20x = 5[x 2 + 4x] = 5x(x + 4)

Thus

as the final correct answer.

 

Example 3:

Simplify .

solution:

The denominator is obviously fully factored. For the numerator, only slight factoring is possible:

3t 2 + 5ts - 7s 2 t = t (3t + 5s - 7s 2 + 4st )

Therefore

as the final simplified form.

 

Example 4:

Simplify .

solution:

First, factoring a 2 + 4a = a (a + 4)

and ax + 4x = x (a + 4)

Thus

as the final simplified result.

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