What You Need to Know About GMAT Exponents
Exponents are among the common algebra concepts that you can expect to encounter on the GMAT quantitative section — primarily on problem solving questions, but occasionally on data sufficiency questions as well.
Definition: An exponent, or power, refers to the number of times a number (called the base) is used as a factor, i.e. the number of times it has been multiplied by itself.
For example, in the number 23, the base is 2 and the exponent is 3. This means that 2 is multiplied by itself three times. Thus, 23 = 2*2*2 = 8.
Foundational GMAT Exponent Rules
The GMAT often tests your ability to manipulate exponents and combine them or separate them in order to simplify an algebraic expression, solve for a variable, combine or simplify polynomials, or simply find a resulting number. Make sure you learn these core exponent rules that will enable you to solve anything the GMAT throws at you involving exponents:
Rule 1: You cannot combine bases or exponents when adding or subtracting terms.
Ex.1: a3 + b3 does not equal (a + b)3
Ex.2: a2 + a3 does not equal a(2 + 3)
Rule 2: You can combine bases when multiplying or dividing terms, provided the exponents are the same. Simply multiply or divide the bases.
Ex.1: a3 * b3 = (a*b)3
Ex.2: a3 / b3 = (a/b)3
Rule 3: You can combine exponents when multiplying or dividing terms, provided the bases are the same. Simply add the exponents in the case of multiplication and subtract the exponents in the case of division.
Ex.1: a2 * a3 = a(2+3) = a5
Ex.2: a5 / a2 = a(5-2) = a3
Rule 4: When raising an exponential number to a power, multiply exponents.
Ex.1: (a2)3 = a(2*3) = a6
Rule 5: Any number raised to the first power equals itself.
Thus, a1 = a
Rule 6: Any number raised to the zero power equals one.
Thus, a0 = 1
Rule 7: With negative exponents, take the inverse of the number and change the negative exponent to a positive one.
Ex.1: 3-2 = 1/32 = 1/9
Ex.2: 1/3-2 = 32 = 9
Exponent Trick: When in doubt, write it out!
“When in doubt, write it out!” is a saying that I use all the time with my students. It’s a helpful reminder that if you’re ever looking at an exponent problem and you can’t remember what the rules say to do, simply write out verbatim what the exponent is telling you.
For example, with something like (a2)3, you may forget the rule and think to yourself, “Shoot! Do I add the exponents? Or multiply them? I forget!” Simply write out what it’s telling you. Technically, it’s telling you to take a2 and multiply it by itself three times. Of course, a2 itself is simply a*a. Thus, the problem is telling you that you’re taking (a*a) and multiplying it by itself three times, or (a*a)(a*a)(a*a) which = a*a*a*a*a*a which is 6 a’s, or a6.
That’s the same outcome as if you remembered the rule, but it’s a fool-proof way to ensure that you don’t make a careless error on test day.
It’s also a useful trick on more difficult exponent problems, as illustrated in this graphic on the right. Many students mistakenly think that (x+y)2 = x2 + y2, but of course if you write out exactly what’s going on, which is that technically you’re multiplying (x+y) by itself, you’ll discover that there’s actually a middle term of 2xy as well and avoid the common trap that the test makers are hoping you fall into.
So remember, “When in doubt, write it out!”
On the GMAT
You can think of the rules above as foundational tools that will enable you to solve more difficult GMAT exponent questions. You may need to be creative and combine several of the rules within the same question to solve it. Consider this example:
Question: If (1/5)m * (1/4)18 = 1/(2*1035), then m =
Give it a try and see how you do. Once you’re done check out this solution video that introduces another advanced GMAT exponent concept about solving for a variable when the variable is in the exponent: