Exponential Functions and Exponent Rules · Sub-skill drill
Fractional and Radical Exponents
A fractional exponent represents a root: x^(1/2) is the square root of x, x^(1/3) is the cube root, x^(2/3) is the cube root of x squared, and so on. SAT questions on this topic test whether you can convert freely between radical and fractional-exponent notation. The fractional form is usually faster for algebraic manipulation; the radical form is more common in geometric contexts. Practice converting in both directions until it is automatic.
How this sub-skill is tested on the SAT
A fractional exponent represents a root: x^(1/2) is the square root of x, x^(1/3) is the cube root, x^(2/3) is the cube root of x squared, and so on. SAT questions on this topic test whether you can convert freely between radical and fractional-exponent notation. The fractional form is usually faster for algebraic manipulation; the radical form is more common in geometric contexts. Practice converting in both directions until it is automatic.
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 6 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.
-
A bacteria population starts at 464 cells and multiplies by a factor of 5 every hour. How many cells are present after 1 hours?
- A 2321
- B 2320
- C 469
- D 464
Worked solution
Answer: B — 2320
Exponential growth: P(t) = P_0 × r^t = 464 × 5^1 = 464 × 5 = 2320 cells.
-
A bacteria population starts at 151 cells and multiplies by a factor of 5 every hour. How many cells are present after 3 hours?
- A 2265
- B 166
- C 18875
- D 3775
Worked solution
Answer: C — 18875
Exponential growth: P(t) = P_0 × r^t = 151 × 5^3 = 151 × 125 = 18875 cells.
-
A bacteria population starts at 639 cells and multiplies by a factor of 5 every hour. How many cells are present after 3 hours?
- A 79875
- B 654
- C 15975
- D 9585
Worked solution
Answer: A — 79875
Exponential growth: P(t) = P_0 × r^t = 639 × 5^3 = 639 × 125 = 79875 cells.
-
A bacteria population starts at 696 cells and multiplies by a factor of 5 every hour. How many cells are present after 3 hours?
- A 17400
- B 711
- C 10440
- D 87000
Worked solution
Answer: D — 87000
Exponential growth: P(t) = P_0 × r^t = 696 × 5^3 = 696 × 125 = 87000 cells.
-
A bacteria population starts at 239 cells and multiplies by a factor of 5 every hour. How many cells are present after 1 hours?
- A 1195
- B 239
- C 244
- D 1196
Worked solution
Answer: A — 1195
Exponential growth: P(t) = P_0 × r^t = 239 × 5^1 = 239 × 5 = 1195 cells.
-
A bacteria population starts at 903 cells and multiplies by a factor of 5 every hour. How many cells are present after 1 hours?
- A 4516
- B 908
- C 903
- D 4515
Worked solution
Answer: D — 4515
Exponential growth: P(t) = P_0 × r^t = 903 × 5^1 = 903 × 5 = 4515 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.