Laws of exponents
Laws of exponents help us to simplify terms containing exponents. We derive these laws here using some good examples
A little reminder before we derive these laws of exponents:
Recall that 2 × 2 × 2 = 2
^{3}
We call 2 the base and 3 the exponent.
Let us now try to perform the following multiplication:
2
^{3} × 2
^{2}
2
^{3} × 2
^{2} = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2
^{5}
Notice that we can get the same answer by adding the exponents
3 + 2 = 5
In the same way,
4
^{3} × 4
^{4} = (4 × 4 × 4) × (4 × 4 × 4 × 4)= 4
^{7}
In general, add exponents to multiply numbers with the same base
Law #1: a
^{n} × a
^{m} = a
^{n + m}
If a stands for any number, a × a × a × a = a
^{4}
By the same token,
If a stands for any number, a × a × a × a × a × a × a = a
^{7}
a
^{4} × a
^{7} = a
^{4 + 7} = a
^{11}
We get
5 × 5 × 5 × 5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
Rewrite the problem:
We get
5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
× 5 × 5 × 5
Notice that
5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
= 1
The reason for this is that whenever you divide something by the same thing, the answer is always 1
The problem becomes 1 × 5 × 5 × 5 = 5 × 5 × 5 = 5
^{3}
Notice that you can get the same answer if you do 8  5 = 3
Let's do also
7^{15}
/
7^{9}
We get
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
Rewrite the problem:
We get
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
× 7 × 7 × 7 × 7 × 7 × 7
Notice Once again that
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
= 1
The reason for this is that whenever you divide something by the same thing, the answer is always 1
The problem becomes 1 × 7 × 7 × 7 × 7 × 7 × 7 = 7
^{6}
Notice that you can get the same answer if you do 15  9 = 6
In general, when dividing with exponents, you can just subtract the exponent of the denominator from the exponent of the numerator.
Law #2:
a^{m}
/
a^{n}
= a
^{m  n}
What about
7^{9}
/
7^{15}
It is the same problem as before. However, this this time 9 is on top and 15 is at the bottom
We can just use the formula
a^{m}
/
a^{n}
= a
^{m  n}
7^{9}
/
7^{15}
= 7
^{9  15} = 7
^{6}
Try now (8
^{3})
^{4}
An important observation:
In (
8^{3})
^{4}, the blue part is the base now and 4 is the exponent
Therefore, you can multiply
8^{3} by itself 4 times.
8^{3} ×
8^{3} ×
8^{3} ×
8^{3} =
8^{3 + 3 + 3 + 3} =
8^{12}
Notice that you can get 12 by multiplying 3 and 4 since 3 × 4 = 12
Law #3: (a
^{n})
^{m} = a
^{n × m}
All the laws of exponents are very useful, especially the last one.
The last makes it easy to simplify (6
^{5})
^{200}
Just multiply 5 and 200 to get 1000 and the answer is 6
^{1000}

Sep 01, 18 04:07 PM
These heart of algebra questions will help you prepare to take the math portion of the SAT
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.