Adding exponents with the same base : GMAT Quantitative Section
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Adding exponents with the same base

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15 Jan 2013, 15:18
I know the usual rules about multiplying exponents and dividing exponents, but I was always under the impression that ADDING exponents with the same base is not possible. For example, I thought it was not possible to simplify $$x^n + x^m$$. But then I saw this operation written and I have no idea how or why this works:

$$3^{17} - 3^{16} + 3^{15} = ?$$
$$= 3^{15}(9 - 3 + 1)$$
$$= 3^{15}(7)$$

And yes, they are equal

Can someone explain how this method works, thanks
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15 Jan 2013, 22:24
hitman5532 wrote:
I know the usual rules about multiplying exponents and dividing exponents, but I was always under the impression that ADDING exponents with the same base is not possible. For example, I thought it was not possible to simplify $$x^n + x^m$$. But then I saw this operation written and I have no idea how or why this works:

$$3^{17} - 3^{16} + 3^{15} = ?$$
$$= 3^{15}(9 - 3 + 1)$$
$$= 3^{15}(7)$$

And yes, they are equal

Can someone explain how this method works, thanks

you are not working with exponents while solving the equation. the solution takes out the exponent part(3^{15) as it is common among all the values and then adds or subtracts the remaining numbers(9, -3, 1).

so all you do while solving such questions is that you find out an exponent-base pair which is common to all the numbers and take it out, leaving behind simple numbers to work with. makes sense??
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15 Jan 2013, 22:35
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hitman5532 wrote:
I know the usual rules about multiplying exponents and dividing exponents, but I was always under the impression that ADDING exponents with the same base is not possible. For example, I thought it was not possible to simplify $$x^n + x^m$$. But then I saw this operation written and I have no idea how or why this works:

$$3^{17} - 3^{16} + 3^{15} = ?$$
$$= 3^{15}(9 - 3 + 1)$$
$$= 3^{15}(7)$$

And yes, they are equal

Can someone explain how this method works, thanks

$$a + b + c = a(\frac{a}{a} + \frac{b}{a} + \frac{c}{a}) = a(1 + \frac{b}{a} + \frac{c}{a})$$

Similarly,

$$3^{17} - 3^{16} + 3^{15} = 3^{15}(\frac{3^{17}}{3^{15}} - \frac{3^{16}}{3^{15}} + \frac{3^{15}}{3^{15}})$$

$$= 3^{15}(3^2 - 3^1 + 1) = 3^{15}(9 - 3 + 1) = 3^{15}(7)$$
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16 Jan 2013, 04:48
hitman5532 wrote:
I know the usual rules about multiplying exponents and dividing exponents, but I was always under the impression that ADDING exponents with the same base is not possible. For example, I thought it was not possible to simplify $$x^n + x^m$$. But then I saw this operation written and I have no idea how or why this works:

$$3^{17} - 3^{16} + 3^{15} = ?$$
$$= 3^{15}(9 - 3 + 1)$$
$$= 3^{15}(7)$$

And yes, they are equal

Can someone explain how this method works, thanks

Check out this post on how to take common terms when dealing with exponents:

http://www.veritasprep.com/blog/2011/07 ... s-applied/
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16 Jan 2013, 13:56
hitman,

There are no general rules for addition or subtraction of exponents(unlike multiplication and addition), instead on the GMAT almost all such problems are simplified using the distributive property of multiplication over addition:
$$a(b+c) = ab + ac$$

For example:
$$42 + 56 = (7)(6) + (7)(8) = 7(6+8) = 7(14) = 98$$

We use the same idea for exponent terms:
$$3^{11} - 3^{10} = (3^{10})(3) - 3^{10}(1) = 3^{10}(3 - 1) = 3^{10}(2)$$

Another example that I use with students:
$$2^{20} - 3(2^{18}) = 2^{18}(2^2 - 3) = 2^{18}(4 - 3) = 2^{18}$$

Here is a list of official GMAT problems that use the same general principle to solve these types of problems:
If you haven't taken the GMATPrep practice test, then skip the ones listed as Official GMATPrep.

1) Old GMAT Paper Test(Easy): 5-12-5-13-a-5-25-b-10-25-c-6-5-12-d-10-12-5-e-49552.html
2) Official GMATPrep(Medium): which-of-the-following-is-equal-to-the-value-of-60927.html
3) Official GMAT Test(Medium): what-is-the-greatest-prime-factor-of-2-100-2-96-a-2-b-70126.html
4) Official GMAT Test(Hard): if-5-x-5-x-3-124-5-y-what-is-y-in-terms-of-x-109080.html
5) Official Guide GMAT 13th Edition(Hard): the-value-of-2-14-2-15-2-16-2-17-5-is-130682.html
6) Official GMATPrep Software(Hard): what-is-the-greatest-prime-factor-of-104757.html
7) Official GMATPrep(Hard): if-3-x-3-x-1-162-then-x-x-1-a-12-b-16-c-20-d-44795.html
8) Official GMAT Test(Medium): if-2-x-2-x-2-x-2-x-2-n-what-is-x-terms-of-n-a-54074.html?fl=similar
9) Official GMATPrep(Hard): http://www.beatthegmat.com/gprep-2-2-2- ... 36087.html ; this can also be solved by using the concept of Geometric series.
10) Official GMATPrep(Hard): if-2-x-2-x-2-3-2-13-what-is-the-value-of-x-130109.html

I believe that is pretty much all of the Official GMAT questions on this subtopic, other than the ones on the real exam over the last few years.

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16 Jan 2013, 16:43
Thank you everyone for your replies. Seeing the multiple takes on methodology gave me a great feel for the concept. It was one of those 'so obvious I missed it' things.
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20 Mar 2014, 07:54
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Re: Adding exponents with the same base   [#permalink] 20 Mar 2014, 07:54
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