Laws of exponentsLaws 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}
Let's do
5^{8}
5^{5}
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} 




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