Ashna K.
asked • 08/29/17what is 10 to the power 10 to the power -10 i.e. (10^10^-10)
I was thinking the answer would be 10^-100, but the answer is 1. Can someone explain how?
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3 Answers By Expert Tutors
Arthur D. answered • 08/29/17
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Mathematics Tutor With a Master's Degree In Mathematics
10^10^(-10)
exponentiation is right associative
this expression becomes:
10^[10^(-10)]
10^[1/(10^10)]
10^[1/10,000,000,000]
you want the ten-billionth root of 10
this is ≈ 1.0000000002302579 which is approximately equal to 1
1.0000000002302579^10,000,000,000=9.99993007628≈10
It does not matter that one exponent was negative. You always go from right to left or from top to bottom. Example: 2^2^2^2=2^(2^[2^2])=2^(2^4)=2^16=65,536. And you need a calculator or a website.
Ashna K.
Thanks much! I am kind of getting it.
So the power property of exponents where you multiply the 2 exponents only works if both exponents are positive?
Since one is positive, and one negative in our case, we have to go right associative first?
BTW: is there any way to figure that the ten-billionth root of 10 is 1.00000.... without a calculator? This was on a test where no calculators are allowed and you do 30 questions in 50 minutes(High School)
Thanks again
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08/29/17
Mark M.
Arthur, clearly explicated, yet would you agree (I think you already did) that (1010)-10 ≠ 1 (and is only in the world of limits.)
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08/29/17
Arthur D.
tutor
It doesn't matter that one of the exponents was negative; you always go from right to left.
Start at the top and go down or start at the right and go left.
2^2^2^2=2^(2^[2^2])=2^(2^4)=2^16=65,536
The answer to BTW is "no." A calculator might not even work. I have a website that I use for calculations like this.
I would have reasoned that 1^10,000,000,000=1 and 2^4=16 which exceeds 10. Therefore the number must be between 1 and 2 and nowhere near the number 2.
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08/29/17
Arthur D.
tutor
Mark, I completely agree with you and a very good point made. I think the problem was given to see if the student understood how to simplify nested exponents.
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08/29/17
Arthur D.
tutor
You're welcome. Ashna. It doesn't matter that one exponent was negative. You always go from right to left or top to bottom.
2^2^2^2=2^{2^([2^2])}=2^{2^4}=2^16=65,536
I don't know of any way except to use a calculator and that might give limited accuracy. I would have reasoned that
1^10,000,000,000=1 and 2^4=16 which exceeds 10; therefore the answer must be between 1 and 2 but nowhere even close to 2.
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08/29/17
Ashna K.
Thanks, got it!
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08/29/17
Mark M. answered • 08/29/17
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Mathematics Teacher - NCLB Highly Qualified
J.R.S.
Your presentation of the law is accurate, yet it is for the multiplication of like bases with exponents.
My reading of the expression is
(1010)-10
In this case a power to a power is the product of the powers
(1010)-10 = 10-100
Ashna K.
The expression is indeed (1010)-10 and I agree that the answer should be 10-100
However, this is a a question from a test and their answer is 1. Actualy, if you put it in a calculator on google, you also get 1.000...... as the answer.
So something I am not able to understand.
Thanks for your help!
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08/29/17
J.R. S. answered • 08/29/17
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Ph.D. in Biochemistry--University Professor--Chemistry Tutor
When multiply powers of ten, you add the exponents. For example,
103 x 104 = 107
Thus, 1010 x 10-10 = 1010-10 = 100 =1
Mark M.
Your statement of the law is correct. Yet it is for the multiplication of like bases with exponents.
As I read the problem the expression is
(1010)-10, if so, a power to a power is the product of the powers.
(1010)-10 = 10-100
Ashna, suggest you request and explanation from the person who gave the answer "1."
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08/29/17
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Mark M.
08/29/17