When evaluating exponent questions in the quantitative section, many test takers freak out when faced with seemingly messy but frequently appearing problems. “Do I need to know logs?” “Wait, is this testing something from Calculus?” Have no fear – while exponents show up often on the GMAT, how these questions show up is repetitive, requiring that we take the same steps over and to solve them. Key things we need to know when it comes to exponents:
When we multiply numbers containing exponents, we add the exponents… but they must have the same base. For example: 2^2 + 2^2 = 2^4. We add the 2 powers to get to 2 raised to the power of 4. But we cannot add 2^2 + 4^2 – the 4 must be converted to the base of a 2. Therefore, 2^2 + (2^2)^2 = 2^2 + 2^4 = 2^6. 2^2 + 3^2 cannot be added just through the exponents. We must simply the number first and then add. Therefore, 2^2 + 3^2 = 4 + 9 = 13.
For the division of numbers containing exponents, we subtract the exponents, and similar rules about bases apply. For example: 2^4/2^2 = 2^(4-2) = 2^2 But we cannot do anything with 3^2/2^2. We simplify to 9/4.
When adding numbers containing exponents, we cannot add the exponents, and same goes for subtraction. For example: 5^4 + 5^2 is not equal to 5^6, but is in fact 750 (625 + 125 = 750). Therefore, when we see a tricky problem with numbers with a series of exponents, then we take a pattern-based approach and not merely a simplification of factors, bases, and exponents. Take this tricky test question: What is the value of 5 + 4 * 5 + 4 * 5^2 + 4 * 5^3 + 4 * 5^4 + 4 * 5^5? This is technically a sum of a geometric sequence question and can be solved using the equation Sn = a(r^n – 1) / r – 1 where Sn is equal to the sum of the terms, a is the first term, and r is the common ratio. But, it is highly unlikely that we are able to remember such a complex equation and plus, the GMAT is definitely not testing our ability to remember formulas. Instead, a better path to take is in the form of factoring. We know we have a sum that incorporates six terms. If all terms were equal to each other, then the sum would be Sn = 6 * (4 * 5^5) = 24 * 5^5 = which would be somewhere around 5^7 if we rounded 24 up to 25 (or 5^2). Then, in assessing the answer choices, we know that the sum must be a bit less than 5^7, leaving us with one – and the correct – option (A) 5^5. And for the final tip around exponents:
Don’t do so much math when it is just about the units digit. We’ll also tackle this concept through a practice problem. What is the units digit of 7^75 + 6? We promise, you never, ever need to multiply out seven 75 times to get to the right answer. A pattern based approach is also appropriate here. When you think about outcomes when 7 is raised to a power: 7^1 = 7 7^2 = 49 7^3 = 343 7^4 = 2401 7^5 = 16807 Telling us that the pattern repeats in a 7-9-3-1. 75/4 gives us a remainder of 3, which means that 7^75 will end in a 3 units digit, same as 7^3. 3 + 6 = 9 giving us the correct answer of (E) 9. When it comes to exponents, remember – you never have to multiply numbers hundreds of times by hand, there is always a simpler way! | 
