Comparing Exponentiation: 20^100 vs. 400^40?

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In summary, the given problem requires the use of exponent laws and mental math to determine which is greater between 20^100 and 400^40 without using a calculator or directly evaluating the expressions. The key to solving this problem lies in recognizing the relationships between the bases and exponents of the two expressions and applying the laws of exponents to simplify the expressions.
  • #1
pandamonium786
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Homework Statement


Without using a calculator and without evaluating expressions, determine which is greater. 20^100 or 400^40

Homework Equations


None, however some exponent laws and possibly mental math are applicable.

The Attempt at a Solution


I don't know how to reach an answer without some type of evaluation; which goes against the question.
 
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  • #2
pandamonium786 said:

Homework Statement


Without using a calculator and without evaluating expressions, determine which is greater. 20^100 or 400^40

Homework Equations


None, however some exponent laws and possibly mental math are applicable.

The Attempt at a Solution


I don't know how to reach an answer without some type of evaluation; which goes against the question.

You should at least put forth some guesses. Hint -- what can you say about how many zeros will be in each answer?
 
  • #3
pandamonium786 said:
without some type of evaluation
Write 400 in terms of 20.
 
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  • #4
berkeman said:
You should at least put forth some guesses. Hint -- what can you say about how many zeros will be in each answer?

I don't know if i am right, but would there be 3 0's in 20^100 and around 4 0's in 400^40?
 
  • #5
Bystander said:
Write 400 in terms of 20.
What do you mean?
 
  • #6
pandamonium786 said:
I don't know if i am right, but would there be 3 0's in 20^100 and around 4 0's in 400^40?

No. Please review how exponents work.

10^2 = 10*10 = 100

10^3 = 10*10*10 = 1000

...and so on... :smile:
 
  • #7
pandamonium786 said:
What do you mean?
Can you relate "400" to "20" or "20" to "400?" Are there some simple operations you can perform on either number to get the other?
 
  • #8
Bystander said:
Can you relate "400" to "20" or "20" to "400?" Are there some simple operations you can perform on either number to get the other?

well 20 x 20 = 400 or 400/20 = 20
 
  • #9
pandamonium786 said:
well 20 x 20 = 400
Give you any ideas?
 
  • #10
Bystander said:
Give you any ideas?

Does that mean that since 20^2 = 400 the 400^40 has to be greater than 20^100. Since 400^2 alone is approximately greater than 20^2.
 
  • #11
pandamonium786 said:
Does that mean that since 20^2 = 400 the 400^40 has to be greater than 20^100. Since 400^2 alone is approximately greater than 20^2.

No, it does not mean that; it is not even true.
 
  • #12
You're playing with the right ideas, but do what Berkeman has recommended regarding laws of exponents; N(k x j) = (Nk)j .
 
  • #13
Poster has been warned to stop asking for the answer and PM berkeman some work so the thread can be reopened
Ray Vickson said:
No, it does not mean that; it is not even true.

then what do you propose the answer is and how would you get to it?
 
  • #14
P
Bystander said:
You're playing with the right ideas, but do what Berkeman has recommended regarding laws of exponents; N(k x j) = (Nk)j .

Please expand. Also how would you reach the answer logically?
 
  • #15
pandamonium786 said:
Please expand
I'll nudge you in the direction you need to go, but you have to do the work.
Take what you did earlier,
pandamonium786 said:
20^2 = 400
pick some values for N, j, k and put them into the general statement I gave you for one of the laws of exponents.
Bystander said:
N(k x j) = (Nk)j
 
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  • #16
Bystander said:
I'll nudge you in the direction you need to go, but you have to do the work.
Take what you did earlier,

pick some values for N, j, k and put them into the general statement I gave you for one of the laws of exponents.

So would 400^40 be greater because
Bystander said:
I'll nudge you in the direction you need to go, but you have to do the work.
Take what you did earlier,

pick some values for N, j, k and put them into the general statement I gave you for one of the laws of exponents.

So that means that 20^100 is actually greater because 20^ 20x20 is bigger than 400^ 20x2.
 
  • #17
Quick tutorial on a couple PF features: the Green bar above the reply box will allow you to insert subscripts, x32n2, or superscripts (exponents), x2 or 20100.
Try redoing what you just did and use this feature --- it might help you visualize just what is "base," (the 20 or the 400) and what is exponent (superscript).
 
  • #18
pandamonium786 said:
then what do you propose the answer is and how would you get to it?

PF Rules forbid me from telling you the answer. We can give hints only. You have been given plenty of helpful hints already.
 
  • #19
pandamonium786 said:

Homework Statement


Without using a calculator and without evaluating expressions, determine which is greater. 20^100 or 400^40

Homework Equations


None, however some exponent laws and possibly mental math are applicable.

The Attempt at a Solution


I don't know how to reach an answer without some type of evaluation; which goes against the question.
I don't believe that you are prohibited from doing every calculation - I think that you don't want you to directly calculate 20400 and 40040. I'm reasonably sure that you are permitted to do some calculations.
 
  • #20
Thread closed temporarily to discuss the rules with the OP...
 

What is the difference between 20^100 and 400^40?

The difference between these two numbers is that they have different bases and exponents. 20^100 has a base of 20 and an exponent of 100, while 400^40 has a base of 400 and an exponent of 40.

Which number is greater, 20^100 or 400^40?

In this case, 400^40 is greater because the base (400) is bigger than the base of 20^100 (20). Even though the exponent of 20^100 is larger, the base has a larger impact on the overall value.

Can you explain why 20^100 is not greater than 400^40?

As mentioned earlier, the base plays a significant role in determining the value of an exponential expression. In this case, even though 20^100 has a larger exponent, the base of 400^40 is much larger, resulting in a higher value.

How do you calculate the value of 20^100 or 400^40?

The value of an exponential expression can be calculated by multiplying the base by itself the number of times indicated by the exponent. For example, 20^100 would be calculated as 20 x 20 x 20 ... (100 times), while 400^40 would be calculated as 400 x 400 x 400 ... (40 times).

Is there a general rule for comparing exponential expressions with different bases and exponents?

Yes, when comparing exponential expressions, the base with the larger value will result in a greater overall value, regardless of the exponent. This is because the base is being multiplied by itself multiple times, and a larger base will result in a larger product.

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