Adding exponents with the same base

Question

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

Moderator Note: Check for more HERE.

Can someone explain how this method works, thanks Check for more HERE .

MacFauz · Accepted Answer

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)

handamanik · Answer

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??

KarishmaB · Answer

Check out this post on how to take common terms when dealing with exponents: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/07 ... s-applied/

dabral · Answer

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): https://gmatclub.com/forum/5-12-5-13-a-5 ... 49552.html
2) Official GMATPrep(Medium): https://gmatclub.com/forum/which-of-the- ... 60927.html
3) Official GMAT Test(Medium): https://gmatclub.com/forum/what-is-the-g ... 70126.html
4) Official GMAT Test(Hard): https://gmatclub.com/forum/if-5-x-5-x-3- ... 09080.html
5) Official Guide GMAT 13th Edition(Hard): https://gmatclub.com/forum/the-value-of- ... 30682.html
6) Official GMATPrep Software(Hard): https://gmatclub.com/forum/what-is-the-g ... 04757.html
7) Official GMATPrep(Hard): https://gmatclub.com/forum/if-3-x-3-x-1- ... 44795.html
8) Official GMAT Test(Medium): https://gmatclub.com/forum/if-2-x-2-x-2- ... fl=similar
9) Official GMATPrep(Hard): https://gmatclub.com/forum/if-2-x-2-x-2- ... 30109.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.

Dabral

hitman5532 · Answer

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.

jiminbang · Answer

So this is how it works. You first have 3^17−3^16+3^15=?
then you get
=3^15(9−3+1) for, you have to get a common factor
then you get 3^15(7)
i you care to simplify.
what about a kodu? eh? or 2. or 3. or 1000000000000000000000000000000000000000000000000000000000000000000
If you like this post, tell me through the kodu button

Bunuel · Answer

EXPONENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number  a  multiplied  n  times can be written as  a^n , where  a  represents the base, the number that is multiplied by itself  n  times and  n  represents the exponent. The exponent indicates how many times to multiple the base,  a , by itself.

Exponents one and zero: 
 a^0=1  Any nonzero number to the power of 0 is 1.
For example:  5^0=1  and  (-3)^0=1 
 • Note: the case of 0^0 is not tested on the GMAT.

a^1=a  Any number to the power 1 is itself.

Powers of zero: 
If the exponent is positive, the power of zero is zero:  0^n = 0 , where  n > 0 .

If the exponent is negative, the power of zero ( 0^n , where  n < 0 ) is undefined, because division by zero is implied.

Powers of one: 
 1^n=1  The integer powers of one are one.

Negative powers: 
 a^{-n}=\frac{1}{a^n}

Powers of minus one: 
If n is an even integer, then  (-1)^n=1 .

If n is an odd integer, then  (-1)^n =-1 .

Operations involving the same exponents: 
Keep the exponent, multiply or divide the bases
 a^n*b^n=(ab)^n

\frac{a^n}{b^n}=(\frac{a}{b})^n

(a^m)^n=a^{mn}

a^m^n=a^{(m^n)}  and not  (a^m)^n  (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)

Operations involving the same bases: 
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
 a^n*a^m=a^{n+m}

\frac{a^n}{a^m}=a^{n-m}

Fraction as power: 
 a^{\frac{1}{n}}=\sqrt {a}

a^{\frac{m}{n}}=\sqrt {a^m}

ROOTS

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

General rules: 
•  \sqrt{x}\sqrt{y}=\sqrt{xy}  and  \frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}} .

•  (\sqrt{x})^n=\sqrt{x^n}

•  x^{\frac{1}{n}}=\sqrt {x}

•  x^{\frac{n}{m}}=\sqrt {x^n}

•  {\sqrt{a}}+{\sqrt{b}}
eq{\sqrt{a+b}}

•  \sqrt{x^2}=|x| , when  x\leq{0} , then  \sqrt{x^2}=-x  and when  x\geq{0} , then  \sqrt{x^2}=x

• When the GMAT provides the square root sign for an even root, such as  \sqrt{x}  or  \sqrt {x} , then the only accepted answer is the positive root.

That is,  \sqrt{25}=5 , NOT +5 or -5. In contrast, the equation  x^2=25  has TWO solutions, +5 and -5.  Even roots have only a positive value on the GMAT.

• Odd roots will have the same sign as the base of the root. For example,  \sqrt {125} =5  and  \sqrt {-64} =-4 .

8. Exponents and Roots of Numbers 
     Theory   
 Exponents and Roots on the GMAT: Tips and hints 
 Cyclicity and the remainders on the GMAT 
     Questions   
 Units digits, exponents, remainders problems 
 Tough and tricky DS exponents and roots questions with detailed solutions 
 Tough and tricky PS exponents and roots questions with detailed solutions 
 Exponents DS Questions 
 Exponents PS Questions 
 Roots DS Questions 
 Roots PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

ccooley · Answer

You've already gotten some good answers, but I want to link this article as well for any curious readers who might find this thread! https://www.manhattanprep.com/gre/blog/ ... exponents/ It's on our GRE blog, but all of the math there should apply to the GMAT as well.

Adding exponents with the same base

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