Exponential Functions and Exponent Rules · Sub-skill drill
Applying Exponent Rules
The core exponent rules are: x^a × x^b = x^(a+b), x^a / x^b = x^(a−b), (x^a)^b = x^(ab), and x^(−a) = 1 / x^a. SAT questions test these with both numeric and algebraic bases. The most common errors are mixing addition with multiplication of exponents (writing x^2 × x^3 = x^6 instead of x^5) and mishandling negative exponents. Slow down on every exponent step and verify the rule by name before applying it.
How this sub-skill is tested on the SAT
The core exponent rules are: x^a × x^b = x^(a+b), x^a / x^b = x^(a−b), (x^a)^b = x^(ab), and x^(−a) = 1 / x^a. SAT questions test these with both numeric and algebraic bases. The most common errors are mixing addition with multiplication of exponents (writing x^2 × x^3 = x^6 instead of x^5) and mishandling negative exponents. Slow down on every exponent step and verify the rule by name before applying it.
This sub-skill sits inside the broader Exponential Functions and Exponent Rules topic, which is part of the College Board's Passport to Advanced Math content domain. Passport to Advanced Math is the SAT's bridge to the kind of algebraic manipulation you will see in a college precalculus or calculus course. The questions are not about memorizing identities — they are about fluency with structure. Can you factor a quadratic by inspection? Can you read the vertex of a parabola off its standard form? Can you simplify a rational expression without losing a domain restriction? Can you translate between f(x), a graph, and a table of values without panicking? Most students who plateau between 650 and 720 plateau here, because the section rewards algebraic intuit
Practice questions in this drill set
Below are 7 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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A bacteria population starts at 514 cells and multiplies by a factor of 2 every hour. How many cells are present after 2 hours?
- A 1028
- B 2056
- C 2057
- D 518
Worked solution
Answer: B — 2056
Exponential growth: P(t) = P_0 × r^t = 514 × 2^2 = 514 × 4 = 2056 cells.
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A bacteria population starts at 265 cells and multiplies by a factor of 2 every hour. How many cells are present after 2 hours?
- A 269
- B 1060
- C 1061
- D 530
Worked solution
Answer: B — 1060
Exponential growth: P(t) = P_0 × r^t = 265 × 2^2 = 265 × 4 = 1060 cells.
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A bacteria population starts at 433 cells and multiplies by a factor of 2 every hour. How many cells are present after 2 hours?
- A 1732
- B 1733
- C 866
- D 437
Worked solution
Answer: A — 1732
Exponential growth: P(t) = P_0 × r^t = 433 × 2^2 = 433 × 4 = 1732 cells.
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A bacteria population starts at 260 cells and multiplies by a factor of 2 every hour. How many cells are present after 1 hours?
- A 521
- B 262
- C 260
- D 520
Worked solution
Answer: D — 520
Exponential growth: P(t) = P_0 × r^t = 260 × 2^1 = 260 × 2 = 520 cells.
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A bacteria population starts at 792 cells and multiplies by a factor of 2 every hour. How many cells are present after 3 hours?
- A 4752
- B 798
- C 6336
- D 3168
Worked solution
Answer: C — 6336
Exponential growth: P(t) = P_0 × r^t = 792 × 2^3 = 792 × 8 = 6336 cells.
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A bacteria population starts at 540 cells and multiplies by a factor of 2 every hour. How many cells are present after 3 hours?
- A 2160
- B 3240
- C 4320
- D 546
Worked solution
Answer: C — 4320
Exponential growth: P(t) = P_0 × r^t = 540 × 2^3 = 540 × 8 = 4320 cells.
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A bacteria population starts at 178 cells and multiplies by a factor of 2 every hour. How many cells are present after 3 hours?
- A 712
- B 1424
- C 1068
- D 184
Worked solution
Answer: B — 1424
Exponential growth: P(t) = P_0 × r^t = 178 × 2^3 = 178 × 8 = 1424 cells.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.