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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:

Re: Adding exponents with the same base [#permalink]

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15 Jan 2013, 23:24

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:

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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Re: Adding exponents with the same base [#permalink]

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15 Jan 2013, 23:35

1

This post received KUDOS

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:

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:

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: \(a(b+c) = ab + ac\)

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.

Re: Adding exponents with the same base [#permalink]

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16 Jan 2013, 17:43

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, 08:54

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17 Aug 2017, 21:42

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