Question 1197176: Is the following simplification correct? Why or why not? Use complete sentences to explain your answer.
2^5 * 2^7 = 4^12
Found 3 solutions by ikleyn, math_tutor2020, josgarithmetic:
Answer by ikleyn(52879)   (Show Source):
You can put this solution on YOUR website! .
The correct simplification is 2^5 * 2^7 = 2^(5+7) = 2^12.
It is fundamentally different from 4^12.
Therefore, the equation in your post is incorrect.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
The general rule is
a^b*a^c = a^(b+c)
The bases are the same. In this case, the base is 'a'
We add the exponents b and c to arrive at a single exponential expression on the right hand side.
For this problem
a = 2
b = 5
c = 7
So it should be
a^b*a^c = a^(b+c)
2^5*2^7 = 2^(5+7) = 2^12
Therefore the equation 2^5*2^7 = 4^12 is false.
The '4' should be a 2.
We can use a calculator to evaluate each expression
2^5*2^7 = 4,096
4^12 = 16,777,216
2^12 = 4,096
This helps show 2^5*2^7 = 2^12 is a true statement.
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If you're wondering how the rule a^b*a^c = a^(b+c) works, then let's break down what 2^5 and 2^7 mean.
2^5 means "multiply 5 copies of the base 2"
2^7 means "multiply 7 copies of the base 2"
We can write it out long hand like this
2^5 = (2*2*2)*2*2
2^7 = (2*2*2)*(2*2*2)*2
The parenthesis are useful to group terms, or we might get lost in a sea of '2's.
Then,
2^5*2^7 = [ 2^5 ] * [ 2^7 ]
2^5*2^7 = [ (2*2*2)*2*2 ] * [ (2*2*2)*(2*2*2)*2 ]
2^5*2^7 = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)
2^5*2^7 = 2^12
In other words,
2^5*2^7 = [ 2^5 ] * [ 2^7 ]
2^5*2^7 = [ 5 copies of '2' multiplied ] * [ 7 copies of '2' multiplied ]
2^5*2^7 = (5+7) copies of '2' multiplied
2^5*2^7 = 12 copies of '2' multiplied
2^5*2^7 = 2^12
This is no means a full formal proof of the rule a^b*a^c = a^(b+c), but it hopefully helps illustrate why the rule works.
Answer by josgarithmetic(39630) (Show Source):
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