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Examples of Exponential Expressions

Zero Rule and Order of Operations

Order of Operations

Equations can result in very different answers, depending on the order in which they are worked.

Equations with the same numbers but different results
6 + (3 · 5) (6 + 3) · 5 Terms in parentheses are worked first.
6 + 15 9 · 5
21 45



'PEMDAS' tells us the necessary order in which to work a problem (the 'order of operations'):

  1. Parentheses (everything inside parentheses first)
  2. Exponents
    (left-pointing arrow work these right to left, or highest exponent to lowest when there is more than one exponent.)
  3. Multiplication & Division
    (right-pointing arrow Work these left to right!)
  4. Addition & Subtraction
    (right-pointing arrow Work these left to right!)

Some people remember: "Please Excuse My Dear Aunt Sally."

the PEMDAS dude with parentheses for eyes, an exponent for the nose, a division symbol and multiplication symbols for a mouth, and plus and minus signs for neck and shoulders.

x0+(3+7)2 =
x0+(10)2 =
1+100 =

Exponents with Like Bases

like bases

(x3)2 · x-2 =
x3*2 · x-2 =
x6 · x-2 =
x6+(-2) =

Power to a Negative Power


(x3)-1 =

Power to a Power

When exponents are 'stacked', as in: 232, they are simplified right-to-left (←), top-to-bottom.

Step 1: 32 = (3)(3) = 9  (Square the exponent before you use it. )
Step 2: 29 = (2)(2)(2)(2)(2)(2)(2)(2)(2) = 512

Note! This is different from (23)2.
In this case, the exponents are multipled.

(23)2 =
2(3)(2) =
26 = (2)(2)(2)(2)(2)(2) = 64

Zero Rule and a Negative

Why -1?

x0 = 1, but in this case The negative sign is outside the parentheses. It could also be wrtten: -1(21x3y2z)0. PEMDAS says first work what's inside the parentheses, then the exponent, then the multiplication.

-(21x3y2z)0 =
-1(21x3y2z)0 =
-1(1) =

-(21x3y2z)0 =

Fractions with Exponential Terms

More about fractions & exponents

5753 can be read: 57 ÷ 53
57 ÷ 53 = (57 )( 5-3)
Why? Dividing by a term with an exponent is the same as multiplying by the term with the negative of its exponent.
57 ÷ 53 = (57 )( 5-3) = 57+(-3) = 54

5753 =
57+(-3) = 54

x4x-3 =
x 4-(-3) = x 4+3 = x7

Here's a video showing another way to think about this
Click the sideways triangle to play the video. The video has sound. (Text transcript >>)

Examples where people may disagree

Which numbers or variables are raised to a power?

Calculators interpret a negative sign as -1, and will display the results shown below.
Some instructors and textbooks may disagree with the example and say the terms below are equivalent. Ask your instructors what they would expect.

  -52 ≠ (-5)2 
(-1)(5)2 ≠ (-1·5)2   
 (-1)(5)(5) ≠ (-1·5)(-1·5)
(-1)(25) ≠ (-5)(-5)   
-25 ≠ 25   

An exponent only applies to the number or variable it is next to. If it is next to parentheses, it applies to everything in the parentheses.

Common Errors: Don't Go There!

In addition & subtraction of like terms, the exponent is unchanged

This example shows different results of adding like terms and multiplying exponents.


x2+x2 ≠ x(2)(2)
42+42 ≠ 4(2)(2)
(4·4) + (4·4) ≠ 44              
16 + 16 ≠ 256   
      32 ≠ 256

x2+x2 = 2(x2) = 2x2.
We can't combine x and x2. Terms with different exponents cannot be combined. x can be thought of as x1.

x2+x2+x =
2x2+x (correct final answer)


Equal to; the same as.
An exponent tells how many times to multiply a number or term (the "base" ). The exponent is called a power. In 63, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6).
Like Bases
The base is the number or term that is raised to a power. Like bases have the same number or term raised to a power. The power (exponent) may be different.
Like Bases:
x3, x4
(ab)2, (ab)y
2x3, x2
NOT Like Bases:
y3, x3
(xy)2, y2
(2x)3, x3
Like Terms
Like terms are terms that have the exact same variables raised to the exact same exponents. A term is a number, variable, or the product of a number and variable(s). These are each terms: 4, x, 4x, 4x2. In an expression, terms are separated by addition or subtraction operators (+,-).
An exponent. A power tells how many times to multiply a number or term. In 63, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6).
Reduce to lowest terms: eliminate parentheses, distribute, combine like terms, get rid of negative exponents.


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This material is based upon work supported by the National Science Foundation (NSF) under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.