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Exponents

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by MartyMurray » Wed Mar 30, 2016 10:29 pm
1) 2^5+2^5+3^5+3^5+3^5 = ?

I am not sure whether for this question you are looking for an exponent answer or the actual answer. So here are both.

First, notice that 2� + 2� = 2 x 2� = 2¹ x 2�

When you have a base raised to a power multiplied by the same base also raised to a power, you just add the exponents.

So 2¹ x 2� = 2�

Now notice that you have 3� + 3� + 3� = 3 x 3� = 3¹ x 3� = 3�

So the expression simplifies to the following.

2� + 3� = 64 + 729 = 793

2) 10^8-10^2/10^7-10^3=?

Start by dividing top and bottom by 10². When you divide a number raised to a power by the same number raised to a power, subtract exponents.

For example, 10� ÷ 10² = 10�.

So (10� - 10²)/(10� - 10³) ÷ 10²/10² = (10� - 1)/(10� - 10) = (1,000,000 - 1)/(10 x (10� - 1)) = 999,999/(10 x 9,999) = 111,111/(10 x 1111) = Some Outrageous Number

Are you sure that you copied this one right. It's not coming out very cleanly?

Anyway, those are some of the principles that you would use to answer GMAT questions based on exponents, and GMAT questions will generally solve to a simple answer.
Last edited by MartyMurray on Thu Mar 31, 2016 4:47 am, edited 1 time in total.
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by DavidG@VeritasPrep » Thu Mar 31, 2016 4:17 am
I think the actual question asks what (10� - 10²)/(10� - 10³) is closest to. The thing to see here is that if the question is asking us what the expression is closest to, it's inviting us to estimate. 10� is an enormous number: 100000000. Subtracting 10² will barely make a dent. Same deal in the denominator. 10� is enormous. 10³ is relatively insignificant. So ignore the smaller terms. If we're going to estimate, we can simply evaluate 10�/10� = 10, which I believe is the answer.
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by MartyMurray » Thu Mar 31, 2016 4:30 am
DavidG@VeritasPrep wrote:I think the actual question asks what (10� - 10²)/(10� - 10³) is closest to.
Ah, what a relief. Some sanity has been brought to the situation.
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by GMATGuruNY » Thu Mar 31, 2016 4:37 am
When you post a problem, please include the answer choices.
2^5+2^5+3^5+3^5+3^5=?

A) 5^6
B) 13^5
C) 2^6+3^6
D) 2^7+3^8
E) 4^5+9^5
An alternate approach is to BALLPARK.

2� = 32.
3� = 3(3�) = 3*81 ≈ 250.

Thus:
2� + 2� + 3� + 3� + 3� = 2(32) + 3(250) ≈ 60+750 ≈ 800.

Since 3� ≈ 250, 3� ≈ 750.
Thus, the only viable answer choice is C.
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by Brent@GMATPrepNow » Thu Mar 31, 2016 6:10 am
Here's the full version of question #2 (with my solution):
The value of (10� - 10²)/(10� - 10³) is closest to which of the following?

a)1
b)10
C)10²
d)10³
e)10�
Since the answer choices are so spread out, we can be quite aggressive with out estimation.

First, 10� - 10² can be rounded to 10�

Is this reasonable? Well, 10� equals 100,000,000, so subtracting 10² (aka 100) from 100,000,000 still leaves us with about 100,000,000

Similarly, 10� - 10³ can be rounded to 10� (10,000,000 - 1000 is pretty close to 10,000,000)

So, by estimating, we get:
(10� - 10²)/(10� - 10³) ≈ (10�)/(10�) = 10¹ = 10 = B
Last edited by Brent@GMATPrepNow on Thu Mar 31, 2016 6:13 am, edited 1 time in total.
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by DavidG@VeritasPrep » Thu Mar 31, 2016 6:12 am
And here are the answer choices for (10� - 10²)/(10� - 10³) is closest to:

A) 1
B) 10
C) 10^2
D) 10^3
E) 10^4
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by DavidG@VeritasPrep » Thu Mar 31, 2016 6:14 am
At least once a week I post something, only to see that Brett has posted the same thing one minute before. :)
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by Brent@GMATPrepNow » Thu Mar 31, 2016 6:15 am
At least once a week I post something, only to see that Davi.... wait a minute!!!!
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by CAREY33 » Thu Mar 31, 2016 9:43 am
Apologies for not being more clear with my question and not posting the possible answers but it looks like you guys figured it out.

I was a little confused whether or not there was any kind of rule for when you add or subtract exponents in these cases. For instance in the first question when trying to add numbers with different bases and then in the second question when trying to subtract numbers with the same base but different exponents.

It appears though the way to tackle these questions is find the value or approximate value for each of the numbers and then add/subtract?
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by DavidG@VeritasPrep » Thu Mar 31, 2016 9:50 am
CAREY33 wrote:Apologies for not being more clear with my question and not posting the possible answers but it looks like you guys figured it out.

I was a little confused whether or not there was any kind of rule for when you add or subtract exponents in these cases. For instance in the first question when trying to add numbers with different bases and then in the second question when trying to subtract numbers with the same base but different exponents.

It appears though the way to tackle these questions is find the value or approximate value for each of the numbers and then add/subtract?
Be careful about over-generalizing. If you see, "closest to" or "approximate" in the question, sure, estimation is great. But other times, it might be most beneficial to factor out the GCF of the two terms. See here: https://www.beatthegmat.com/what-is-the- ... 53058.html
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by regor60 » Fri Apr 01, 2016 8:53 am
Another way is to factor:

10^2(10^6 - 1)/10^3(10^4 - 1)

= (10^3+1)(10^3-1)/10(10^2+1)(10^2-1)

The one's are irrelevant, obviously, so this reduces to 10^6/10^5 or 10
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by CAREY33 » Fri Apr 01, 2016 9:06 am
Thanks for the response.


I see how you did the factoring, but could you explain how you got to: (10^3+1)(10^3-1)/10(10^2+1)(10^2-1)

Im a little confused
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by regor60 » Fri Apr 01, 2016 9:41 am
CAREY33 wrote:Thanks for the response.


I see how you did the factoring, but could you explain how you got to: (10^3+1)(10^3-1)/10(10^2+1)(10^2-1)

Im a little confused
It's called difference of squares. For example, A^2 - B^2 can be factored into (A+B)*(A-B)

10^6-1 is a difference of squares. 10^6 is the square of 10^3 and 1 is the square of, wait for it...1.

Similarly 10^4-1
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