## Examples of Exponential Expressions

## Order of Operations

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

6 + (3 · 5) | (6 + 3) · 5 | Terms in parentheses are worked first. |

6 + 15 | 9 · 5 | |

21 | 45 |

### PEMDAS

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

- Parentheses (everything inside parentheses first)
- Exponents

( work these right to left, or highest exponent to lowest when there is more than one exponent.) - Multiplication & Division

( Work these left to right!) - Addition & Subtraction

( Work these left to right!)

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

`x ^{0}+(3+7)^{2} =`

x^{0}+(10)^{2} =

1+100 =

101

`(x ^{3})^{2} · x^{-2} =`

x^{3*2} · x^{-2} =

x^{6} · x^{-2} =

x^{6+(-2) }=

x^{4}

`(x ^{3})^{-1} =`

x^{(3)(-1)}=

x^{-3}=

1x^{3}

When exponents are 'stacked', as in: 2^{3}^{2}, they are simplified right-to-left (←), top-to-bottom.

`2 ^{3}^{2 }`

Step 1: 3^{2} = (3)(3) = 9 (Square the exponent before you use it. )

Step 2: 2^{9} = (2)(2)(2)(2)(2)(2)(2)(2)(2) = 512

Note! This is different from (2^{3})^{2}.

In this case, the exponents are multipled.

`
(2 ^{3})^{2} = `

2^{(3)(2)} =

2^{6} = (2)(2)(2)(2)(2)(2) = 64

#### Why -1?

x^{0} = 1, but in this case **The negative sign is outside the parentheses**. It could also be wrtten: -1(21x^{3}y^{2}z)^{0}. PEMDAS says first work what's inside the parentheses, then the exponent, **then** the multiplication.

-(21x^{3}y^{2}z)^{0} =

-1(21x^{3}y^{2}z)^{0} =

-1(1) =

-1

`
-(21x ^{3}y^{2}z)^{0} = `

-1

#### More about fractions & exponents

5^{7}5^{3} can be read:
5^{7} ÷ 5^{3}

**5 ^{7} ÷ 5^{3} = (5^{7} )( 5^{-3})**

Why? Dividing by a term with an exponent is the same as multiplying by the term with the

**negative**of its exponent.

**5**

^{7}÷ 5^{3}= (5^{7})( 5^{-3}) = 5^{7+(-3)}= 5^{4}`5 ^{7}5^{3} = `

5^{7+(-3)} = 5^{4}

`x ^{4}x^{-3} = `

x ^{4-(-3)} = x ^{4+3} = x^{7}

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

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.

`
-5 ^{2} ≠ (-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!

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

`x ^{2}+x^{2}`

x^{2}+x^{2} ≠ x^{(2)(2)}

4^{2}+4^{2} ≠ 4^{(2)(2)}

(4·4) + (4·4) ≠ 4^{4}

16 + 16 ≠ 256

32 ≠ 256

**x ^{2}+x^{2} = 2(x^{2}) = 2x^{2}. **

We can't combine x and x

^{2}. Terms with different exponents cannot be combined. x can be thought of as x

^{1}.

`simplify:
x ^{2}+x^{2}+x =
2x^{2}+x (correct final answer)`