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Huzefa Kapadia, an 800-scoring math specialist, breaks SAT Math into fast, high-leverage strategies and clear walkthroughs of the trickier question types.
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Math · Checkpoint 18 of 30 · Advanced Math · ~14 min
Exponents, radicals & nonlinear equations
Exponent rules, exponential growth, radical and rational equations.
The 30-second version
exponential models are y = a·bˣ — a is the start, b is the multiplier per step (b = 1.05 means +5% each period; b = 0.8 means −20%). Exponent rules: multiply → add exponents, power → multiply them, negative → reciprocal, fraction → root. Radical and rational equations have one extra duty: check your answer in the original — squaring both sides can invent fake solutions.
Learn it
Exponent rules you'll actually use
- x³ · x⁴ = x⁷ (same base multiplied → add)
- (x³)⁴ = x¹² (power of a power → multiply)
- x⁻² = 1/x² and x^(1/2) = √x, so x^(3/2) = (√x)³
- x⁰ = 1 for any nonzero x
SAT versions usually ask you to rewrite one expression as another — "which is equivalent to…" — so fluency converting between radical and fractional-exponent form matters.
Exponential growth and decay
"A population of 500 grows 3% per year" → P = 500(1.03)ᵗ. "A car worth $24,000 loses 15% per year" → V = 24000(0.85)ᵗ. The interpretation questions mirror linear ones: a is the initial amount, b carries the percent. Linear = adds the same amount each step; exponential = multiplies by the same factor. "Which model fits?" questions test exactly that distinction.
Radical and rational equations
√(2x + 3) = x → square: 2x + 3 = x² → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0 → x = 3 or −1. Now check: √9 = 3 ✓, but √1 = 1 ≠ −1 ✗. So the only solution is 3. That rejected −1 is called extraneous, and the SAT writes answer choices specifically to catch students who skip the check. For rational equations, multiply through by the denominator — and reject any "solution" that makes a denominator zero.
Common traps
- 15% decay is ×0.85, not ×0.15. The multiplier is what remains.
- Adding exponents across different bases. x³·y⁴ simplifies to nothing.
- Skipping the extraneous check — the wrong answer literally sits in the choices waiting.
Try it
A lab culture starts at 200 cells and doubles every hour. Which expression gives the count after t hours?
Doubling is repeated multiplication: start 200, multiply by 2 each hour → 200·2ᵗ. A is a linear model (adds 2 per hour — a very slow culture). C explodes incorrectly: at t = 1 it gives 400 by luck, but at t = 2 it gives 80,000 instead of 800. Sanity-check models by plugging in t = 1 and t = 2.
Free resources for this checkpoint
Khan Academy Official Digital SAT Prep (opens in a new tab)
Free structured lessons, video instruction, and practice exercises for both SAT sections, built with College Board as the official practice partner. After a Bluebook practice test, you can link your results to get personalized practice recommendations.
Official Practice Partner
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Practice next
Question Bank filter: Math → Advanced Math → Equivalent expressions and Nonlinear equations. Verify current topic scope against the official materials as you drill — the Question Bank's own filters are the authoritative list.
Practice these skillsOriginal Digital SAT-style questions on what this checkpoint covers.