Simplify, Cancel, and Other Math Lingo With Multiple
Meanings
Lisa S. Yocco
Assistant Professor
Department of Mathematical Sciences
Georgia Southern University
AMATYC
November 13, 2009
DEFINTIONS OF “SIMPLIFY”

If an expression A in one variable is simplified to an expression B, then every real number (for
which both expressions are defined) is a solution of the equation A = B. That is, the equation A
= B is an identity.
(Lehmann, Elementary Algebra)

To simplify means to work the problem until nothing else can be done. For instance, 2pi can
be simplified to 6.283 but no further.
(Answers.com)

Simplify (mathematics), to convert a mathematical expression, such as a fraction or
equation, to a simpler form by removing common factors or regrouping terms.
(Encarta.com)
In the process of simplifying an expression, we must make sure that the new simpler expression is
mathematically equivalent to the original expression.
Simplifying Expressions Involving Exponents
1. No parentheses should remain
2. Each variable or constant should appear as a base as few times as possible.
3. Each numerical expression should be calculated.
4. All fractions should be simplified.
5. No negative exponents should remain.
256 x 2 y3
324 x 5 y
divide out any factors common to the numerator
and denominator
3 x 2 4 x5
multiply the constants and use a Rule of
Exponents to multiply the x’s
(4 y)3
rewrite the expression with no negative
exponents and carry out any exponential
evaluations that are possible
x2  y 1
Use the definition of negative exponent to rewrite
the expression as a difference of two fractions,
then get a common denominator to perform the
subtraction
Simplifying Radical Expressions
1. Any indicated multiplications should be performed.
2. Any like radicals should be combined.
3. For any radical with index n, no nth power factors should remain under the radical.
4. The index of the radical should be as small as possible.
5. No radicand should contain a fraction.
6. No radical should appear in a denominator.
7. An expression should contain as few radicals as possible.
3 x

3
x 6

3
x
or
The radicals have to be changed to exponential form so that they can be multiplied
3
4
72x6 y3 z 7
remove all perfect square factors from the
radicand
rewrite the radical so that no fraction remains
under the radical
3
3 8  2 18  4 2
By first simplifying the radicals then combining
like radicals
Simplify the quotient of two
27z 7
remove all perfect cube factors from the radicand
Simplify a complex fraction
by multiplying
3  4i
x
complex numbers
by
the numerator
1
5  2i
and
y
multiplying the numerator and
denominator by
denominator by an expression
x 1 the least

that will make the denominator a
2
y
x common
rational number. Generally this
denominator of
expression is the conjugate of
the individual
the denominator.
fractions
Simplify a complex fraction
25
x
5 1 by changing the

xy y
problem to a division problem
25  5 1 
  
x  xy y 
14  2(12)
2(5 y  6 z )  4(2 y  2 z )
15 x 4  25 x3
3(3)  52
use the Distributive Property to
remove parentheses, then
combine like terms
5x2
follow the order of operations to
evaluate the expression
factor a common factor from
the numerator that will divide
into the denominator
4  x2
x2  x  6
logarithmic expressions
Factor the numerator and
2 log x  log( x  1) using
denominator, then divide out a
factor with its opposite to obtain - logarithmic properties
1
trigonometric expression
sin 60  sin 20
14x2 + 9x2 − 3x2
by combining like terms
trigonometric expression
1  2cos 2 y
using identities
1  2cos y sin y
4 x  3y
y  3x
By multiplying numerator and
denominator by the conjugate
of the denominator
using the trigonometric identity
A B
A B
sin A  sin B  2sin
cos
2
2
Students often confuse solving an equation with simplifying an expression
Solve
3 6

4 x
vs
Properties of equality hold, and we
can multiply both sides by 4x
A
complex, etc.
Simplify
3 6

4 x
We must get a common denominator, 4x, and get
equivalent fractions with the same denominator to
combine them
refers to a specific value, which could be any type of number: real, irrational,
In English, a number of items means more than one, and probably an integer. Such as “I have a
number of students who have poor algebra backgrounds.”
Number
odd, even, prime, composite, positive, negative, whole,
rational, decimal, complex, irrational, natural, imaginary,
nonzero, finite, infinite
.3131131113…
0 e2 one
three-fourths
3
3
p 120
cos
2
4
q
A
and a
4
5

2
444
nineteen
ei a + bi
5/6  3 40
1.234
sin(2x)
can be a
can be a
EQUATION
An equation is a mathematical sentence stating
that one expression equals another. One solves
an equation.
2x + 6 = −7 open equation
5=2–7
false equation
2x + y = 3 linear equation
X2 −4x +5 = 0 quadratic equation
EXPRESSION
An expression is a mathematical phrase. One
simplifies an expression.
2x + 6
X2 −4x +5
2–7
Equivalent expressions have the same value for
every possible value of the variables they include.
3(x-4) and 3x - 12 are equivalent expressions
because no matter what the value of x, they are
equal in value.
Equivalent equations are ones such that the truth
of one implies and is implied by the truth of the
other.
(x−4)2 = 0 and x = 4 are equivalent equations,
even though the expressions in the equations are
not equivalent.
“Equal” is often used carelessly by students.
Consider a calculus student who is asked to
find the maximum of the function
Some traps for students…..
It is true that (−1)2 = (1)2, so
“canceling the squares” gives
−1 = 1
y  2 x 2  4 x  1
and technically gets the correct answer:
y  2 x 2  4 x  1  4 x  4  1  3
x = 2 could mean
OR……………
plot the point on a number line
Draw the vertical line in the rectangular
coordinate axes
y
5
4
3
-3 -2 -1
2
0
1
2
3
1
-5
-4
-3
-2
-1
1
2
3
-1
4
5
x
-2
-3
-4
-5
We evaluate formulas, simplify algebraic expressions, solve equations, cancel common factors, graph
equations, minimize functions, maximize functions, substitute equivalent expressions, expand
expressions, shift graphs, shrink graphs, reflect graphs, stretch graphs, translate graphs, combine like
terms, find inverses of numbers, of functions, of matrices, cross-multiply proportions, satisfy
equations, perform long division, divide synthetically, reduce fractions, reduce matrices, rationalize
denominators, verify identities.
There are maximum values, minimum values, extreme values, critical values, absolute
value, extraneous values.
Identity Element for Multiplication, 1
Identity Element for Addition, 0
1 0 
Identity Matrix, 

 0 1
Identity Function, y = x
Identity sin2x + cos2x = 1
Inverse of a Number (Multiplicative)
Inverse of a Number (Additive)
Inverse Proportion
Inverse Variation
Inverse Function
Inverse Trigonometric Functions
Inverse of a Conditional
Inverse of an Operation
Inverse of a Matrix
Infinitesimal
Infinity
Infinite
A hypothetical number that is
A "number" which indicates a
larger than zero but smaller than quantity, size, or magnitude that
any positive real number.
is larger than any real number.
Describes a set that is not finite
Solve a linear equation
Solve a quadratic equation
Solve the triangle
2 x  9  3x  5( x  7)  3
3x 2  13x  10  0
Remove parentheses, get all
variable terms on one side and
all constant terms on the other
side.
Factor and set each factor equal
to 0.
2 x  9  3x  5( x  7)  3
5 x  9  5 x  35  3
5 x  5 x  38  9
10 x  47
5
5
3x  13x  10  0
2
(3x  2)( x  5)  0
3x  2  0 x  5  0
x
2
3
x  5
5 2  52  x 2
50  x 2
and angles = 45o
5 2  x
x  4.7
From “Math
Spoken Here”
The Number
Any number multiplied by 0 is 0. Any number added to 0
is the number itself. When 0 is subtracted from the
number the number remains the same. Any number
divided by 0 is undefined, except 0 itself, and 0 divided
by 0 is indeterminate. As a digit, 0 is used as a
placeholder in place value systems. 0 is neither positive
nor negative, but it is even. The square root of 0 is 0, and
0 squared is 0.
Factor
Factor 40 means rewrite 40 as a product
Factor x2 – 9 means find two
of primes
polynomials whose product is x2 − 9
Students often confuse “factor” with “solve”
Factor
Solve
x  5x  14
( x  7)( x  2)
2
x 2  5x  14  0
( x  7)( x  2)  0
x  7 x  2
Some improper uses of Math Lingo
loge e x  x
The e’s
8log8 y  y
The 8’s
x2  x
x2  x
x 1
The square and the
square root
Divide both sides by x
and the x’s