From Zero to Hero
(This is one of a series of GMAT tips that we offer on our blog.)
The number 0 is a tricky one on the GMAT, as its unique properties are often either the key to unlocking a difficult solution, or the trap in to which a seemingly logical solution can lead you. Learning the properties of zero (keep in mind that it is an even number) is an important skill, particularly on data sufficiency problems. Even more importantly, never forget to consider zero as a potential value for a variable, as it often produces surprising results. Consider the case of zero as an exponent:
x^0 is, by definition, equal to 1. This concept may have bothered you in high school, as it seems almost arbitrary that an exponent without value would automatically lead to a value of 1. But noting the properties of exponents can help you to prove and more easily embrace and remember this useful device: take, for example, the expression x^2 * x^-2. You could rearrange this two ways:
a) (x^2) / (x^2) –> The negative exponent simply moves that term to the denominator
b) x^(2-2), or x^0 –> When multiplying terms with the same base, taken to different exponents, you simply add the exponents
Because we can prove that (x^2) / (x^2) must be equal to 1, and that the two expressions above are equal to each other, we can prove that x^0 = 1.
Now here comes the payoff – because x^0 is equal to 1, it’s the ultimate in cop-out solutions to difficult problems. Say that a question asks:
For what value of x will 5^x be a factor of 2^10?
2^10 is not divisible by 5 (its only prime factor is 2), but the question might seem to require you to multiply that value out, as well as some potential values of 5^x, in futility to prove that point. However, if 0 is an option, it will set the term equal to 1, which is a factor of any integer, and your work is already done.
Note that the use of 0 as an exponent is always quick and often the only solution to an exponent/divisibility problem. Keeping this device in mind will save you time on the GMAT, as well as enable you to solve some fairly difficult problems.