Exponents and roots questions test a short list of rules, applied to numbers too large to calculate directly. That is the point: GMAT Focus Quant has no calculator, so the question is designed to reward manipulation over arithmetic.
Your first move is almost always the same: rewrite everything with prime bases. 8 is 2³, 27 is 3³, 12 is 2² × 3, and 0.25 is 2−2. Once the bases match, you can compare or equate exponents directly, and expressions such as 4x × 8y collapse into a single power of 2. When a sum of powers appears, such as 3n+2 − 3n, factor out the smallest power: 3n(9 − 1).
For roots, simplify by pulling out perfect squares (√72 = 6√2) and rationalise denominators by multiplying by a conjugate. Remember that √(x²) equals |x|, not x, and that the even root of a negative number is not a real number.
The traps are sign and size. A negative base raised to an even power is positive. A fraction between 0 and 1 shrinks when raised to a higher power, and its square root is larger than the fraction itself. In Data Sufficiency, test 0, 1, −1 and a fraction before trusting a pattern.
4 practice questions
Problem SolvingDivisibility/Multiples/FactorsExponents/PowersMedium-hard
Question 1
What is the greatest integer $k$ for which $\frac{24!}{12^k}$ is an integer?
A$8$
B$10$
C$11$
D$12$
E$22$
Show answer & explanation
Correct answer: B
Key Concept or Approach
$12^k = 2^{2k} \cdot 3^k$. Count the factors of 2 and 3 in $24!$; the scarcer one (after adjusting for the exponent 2) limits $k$.
Step-by-Step Solution
Step 1: Factors of 2 in $24!$: $12 + 6 + 3 + 1 = 22$, so $2k \le 22$ and $k \le 11$.
Step 2: Factors of 3 in $24!$: $8 + 2 = 10$, so $k \le 10$.
Step 3: The greatest $k$ is $10$.
Final Answer
Answer: B
Common trap: 11 - only checking the powers of 2 because 12 contains 2 squared.
Data SufficiencyExponents/PowersMedium-hard
Question 2
If $x$ is an integer, what is the value of $x$?
(1) $x^2 = 9$
(2) $4^x \cdot 2^x = 512$
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: B
Key Concept or Approach
Rewrite everything in base 2: $4^x = 2^{2x}$ and $512 = 2^9$. An even power leaves a sign ambiguity.
Step-by-Step Solution
Step 1: Statement (1): $x = 3$ or $x = -3$. Not sufficient.
Step 2: Statement (2): $2^{2x} \cdot 2^x = 2^{3x} = 2^9$, so $3x = 9$ and $x = 3$. Sufficient.
Step 3: Only statement (2) is sufficient.
Final Answer
Answer: B
Common trap: D: students forget the negative root in (1) and take $x = 3$.
Problem SolvingExponents/PowersMedium-hard
Question 3
If $2^x \cdot 4^{x+1} = 32^{x-1}$, what is the value of $x$?
A$\frac{3}{2}$
B$\frac{5}{2}$
C$3$
D$\frac{7}{2}$
E$7$
Show answer & explanation
Correct answer: D
Key Concept or Approach
Rewrite everything as a power of 2 and equate exponents.
Step-by-Step Solution
Step 1: Left side: $2^x \cdot 2^{2x+2} = 2^{3x+2}$. Right side: $2^{5x-5}$.
Step 2: Set $3x + 2 = 5x - 5$, so $2x = 7$.
Step 3: $x = \frac{7}{2}$.
Final Answer
Answer: D
Common trap: 3 - writing 4^(x+1) as 2^(2x+1) (not distributing the 2), which gives 3x+1 = 5x-5.
Problem SolvingAlgebraRootsHard
Question 4
If $x + \frac{1}{x} = 3$ and $x > 1$, what is the value of $x^2 - \frac{1}{x^2}$?
A$\sqrt{5}$
B$3\sqrt{5}$
C$7$
D$4\sqrt{5}$
E$9$
Show answer & explanation
Correct answer: B
Key Concept or Approach
Factor as a difference of squares: $x^2 - \frac{1}{x^2} = (x+\frac{1}{x})(x-\frac{1}{x})$.
