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Exponents and Their Laws

A basic tutorial on exponents
by

David Huang

on 7 June 2015

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Transcript of Exponents and Their Laws

Exponents
Exponents are basically a way to simplify repeated multiplication. Exponents, along with a base, are called powers.
What are exponents?
5
Powers of 10 and the Zero Exponent
Order of Operations with Powers
Laws of Exponents 1
2
This term is known as a power. In this case, we say "5 to the power of 2. The "5" is called a base and the small "2" is known as an exponent. The exponent tells you how many times to multiply the base by itself. Together, they represent the equation 5 x 5. Thus, 5 would equal 5 x 5 x 5, 5 would be 5 x 5 x 5 x 5 x 5 x 5 x 5 and...well, you get the idea.
Square Numbers
Cubed Numbers
Any number to the power of 2 is a square number. For example, 5 to the power of 2 is a square number because you can create a square with length 5 units and width 5 units.
5 to the power of 3 is 5x5x5. This term is called a cubed number. That's because you can create a cube with length 5 units, width 5 units and height 5 units.
(-4)
3
Simply, this equation means (-4)x(-4)x(-4), which is -64.
-(-4)
3
This is a little different. It means that the outcome of negative four cubed must have another negative sign. Negative four cubed is negative 64. Two negatives make a positive, so negative negative four cubed is 64.
-4
3
In this case, the negative sign and four are separate. So, after you calculate 4 cubed (4x4x4), you add a negative sign. Thus, the answer is negative 64.
2) a) Express the number of unit squares in the large square as a power
b) Express the number of unit cubes in the large cube as a power
3) a) What is the base of this power? What is the exponent?
4) Write these mathematical sentences as powers
a) 8 x 8 x 8 x 8 x 8 = 8
b) -(-5)(-5)(-5) = -(-5)
c) (-4)(-4)(-4)(-4) = (-4)
d) 6 x 6 = 6
1) Write in standard form (as a number).
a) -6 = -36
2
b) 8 = 4096
4
c) -(-3) =243
5
Any base to the power of 0 is always 1.
593859 = 1
0
But in some special cases, the answer is a tiny bit different.
-(-6) = -1
(-3) = 1
0
0
-4 = -1
0
Why so? Because the outcome of (-6) , which is one, has to have another negative sign. Same goes for -4 . The negative sign and 4 are separate. Thus, you take the result of four to the power of zero (one) and add a negative sign.
0
0
Powers of 10
5+7=
10
100
1000
10000
100000
1000000
Number
Power
10
10
10
10
10
10
1
2
3
4
5
6
Powers of 10 are easy to express in exponential form. Notice how in the chart, the exponent of each power represents how many zeros are in the actual number.
But what about numbers that aren't multiples of 10? How do you express them as powers of 10?

3451 = There are 3 thousands, 4 hundreds, 5 tens, and one one. Now we can turn this into a mathematical sentence with powers of 10.

3451 = (3 x 10 ) + (4 x 10 ) + (5 x 10 ) + (1 x 10 )

And that's it! You separate the digits and convert them to powers of 10.
3
2
1
0
There's a certain order in which we do mathematical equations. Here's how to remember it.

BEDMAS
B = Brackets, E = Exponents, D = Division, M = Multiplication, A = Addition, S = Subtraction

Do equations in the order of BEDMAS. However, note that division and multiplication are done in the order that they appear. Addition and subtraction are also likewise.
1. Express each of these powers in standard form
a) 14 =1 b) 956 = 1 c) (-1) = 1 d) -(-1) = -1

2. Write each number in standard form.
a) 2 x 10 =200
b) (3 x 10 ) + (4 x 10 ) + (3 x 10 ) = 3000 + 400 + 30 = 3430
c) (4 x 10 ) + (6 x 10 ) + (5 x 10 ) = 400000 + 60000 + 5 = 460005

3. Express each number as powers of 10.
a) 10000 = 10
b) 3456 = (3 x 10 ) + (4 x 10 ) + (5 x 10 ) + (6 x 10 )
c) 478 = (4 x 10 ) + (7 x 10 ) + (8 x 10 )
2
Practice The Order of Operations
When it comes to solving exponents, sometimes there are shortcuts. For example....

6 ÷ 6
7
3
How come? Because this equation is equal to
6 x 6 x 6 x 6 x 6 x 6 x 6
_________________________
6 x 6 x 6
If you simplify the equation and cross out three sixes on both sides, you will be left with four sixes, which is 6
4
But be careful - this trick only works when all the bases are the same and it is a division statement
This is equal to 6 , which is equal to 6
It turns out there's also a shortcut for multiplying a power several times
6 x
2
6 = 6
4
6
6 = 6
Proof that this works:
The equation is equal to (6 x 6) x (6 x 6 x 6 x 6), which is 6
6
This trick will not work if the bases are different or it is not a multiplication statement.
Laws of Exponents 2
There's another trick that you can use when you're working with exponents. This strategy can only be used when a power is raised to a power
(4 )
2
5
4
10
(4 ) is equal to 4 x 4 x 4 x 4 x 4 . That's equal to 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 , which means that it has an overall exponent of 10.
2
5
2
2
2
2
2
Other Laws of Exponents

(ab) = a x b i.e ((-4) x (-5)) = (-4) x (-5)
c
c
c
(a/b) = a /b
c
c
c
2 x 5
= 4
10
(5/6) = 5/6 x 5/6 x 5/6 = 125/216
3
i.e
by David Huang
3
7
6
4
2
5
3
7
6 is the base, 7 is the exponent
5
3
4
2
0
0
0
3
0
2
1
5
4
0
2
1
0
1. Solve these statements.
a) 5 + (4 + 6) = 25 + 10 = 35
b) 6 x 7 - 8 ÷ 2 = 40
c) (5-1) - (89 ) = 15
2
2. Put brackets in the right place to get the correct answer.
a) 4 + 8 x 2 x 2 = 68
b) 8 ÷ 2 - 4 + 5 = 21
c) 3 - 8 x 2 = 76
2
2
0
2
( )
2
( )
3
2
3) Mary and Joe had to solve a math question.
7 x 9 - (4 x 2)
2
2
Mary got 503. Joe got 3905. Who is right?

7-3
4
1) Write each quotient as a single power
a) 6 ÷ 6 = 6
b) 5 ÷ 5 = 5
c) (-4) ÷ (-4) = (-4)
2) Write each product as a power
a) 7 x 7 = 7
b) 3 x 3 = 3
c) 5 x 5 = 5

3) Simplify, then evaluate.
10 x 10 + 10 = 10000 + 10000 = 20000
2
5
6
2
2
4
1
2
3
4
6
5
3
3
4
6
9
7
2
2
4
1.
Write as a power
a) (6 ) = 6 b) -(2 ) = -2 c) (7 ) = 7

2
3
6
3
4
12
0
6
0
2. Write each expression as a quotient of powers
a) (5 ÷ 6) = 5 ÷ 6 b) ((-13) ÷ (-2)) = (-13) ÷ (-2)
c) (1/3) = 1 ÷ 3

3. Simplify each expression.
a) (4 x 4 ) = 4 b) (12 ) ÷ (12 ) = 12

3
3
3
3
3
3
5
5
5
6
6
6
2
3
4
20
4
5
3
2
14
4
3
2
1
0
2+4
6
( )
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