GRE Quant · Study & Practice

Exponents & Roots

AreaArithmetic DifficultyMedium GRE weightageHigh — exponent rules underpin algebra, scientific notation, growth problems and many Quantitative Comparisons

Exponents are shorthand for repeated multiplication, and almost every question in this chapter reduces to a handful of laws applied carefully. The trouble is never the rules themselves — it is the edge cases the GRE loves: a negative base raised to an even power, a fractional exponent that is really a root, or a Quantitative Comparison where the answer flips depending on whether the base is between 0 and 1. Raising a proper fraction to a higher power makes it smaller, and raising a number between −1 and 0 can change its sign — the exact places the test sets its traps. Roots are simply the reverse operation, and the GRE keeps them at high-school level: square roots, cube roots, and simplifying radicals, with no need for a scientific calculator (the on-screen one is basic and won’t even take a root of a non-perfect square cleanly). This chapter drills the laws of exponents, negative and fractional powers, square and cube roots, simplifying radicals, and scientific notation — the compact skill set behind a surprising share of the section.

Topics

1 Laws of Exponents 4 worked · 5 practice · 8-Q test Start → 2 Negative & Fractional Exponents 4 worked · 5 practice · 8-Q test Start → 3 Square & Cube Roots 4 worked · 5 practice · 8-Q test Start → 4 Simplifying Radicals 4 worked · 5 practice · 8-Q test Start → 5 Scientific Notation 4 worked · 5 practice · 8-Q test Start →

⚡ GRE shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Rewrite everything to a common prime base (8 = 2³, 27 = 3³, 25 = 5²). Equal bases let you add, subtract or match exponents directly.
  • Fractional exponent a^(m/n): take the n-th ROOT first (keeps numbers small), then raise to the m-th power.
  • Negative exponent = reciprocal, nothing more. It never changes the sign of a positive base.
  • Memorise squares to 15² = 225 and cubes to 5³ = 125 — the GRE builds root questions around them so you skip the calculator.
  • Simplify a radical by pulling out the biggest perfect-square factor: √72 = √(36·2) = 6√2.
  • In scientific notation, compare powers of 10 first; only look at coefficients when the powers are equal.
  • For a base between 0 and 1, higher powers SHRINK it and negative powers GROW it — the opposite of your intuition.

⚠️ Common mistakes & traps

GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Reading −2⁴ as (−2)⁴. The first is −16, the second is +16; the exponent binds tighter than the sign.
  • Splitting a root over a sum: √(a + b) ≠ √a + √b (√25 = 5, not 3 + 4).
  • Thinking a negative exponent makes the value negative — it only takes the reciprocal.
  • Adding exponents when the BASES differ (a²·b³ does not combine).
  • Assuming √x < x always; for 0 < x < 1 the square root is larger than x.
  • Forgetting to renormalise scientific notation so the coefficient stays between 1 and 10.
  • Treating √a + √b as combinable when the radicands differ (3√2 + 5√3 cannot be merged).

📈 GRE exam insight & question patterns

Quantitative Comparison — 0 < x < 1. Quantity A: x². Quantity B: x³. Which is greater?

Quantity A. For a proper fraction, each higher power is smaller, so x² > x³.

Same base — solve 4ˣ = 8.

Write both in base 2: 2^(2x) = 2³, so 2x = 3 and x = 3/2.

Numeric Entry — simplify √180 to the form a√b with b square-free; enter a.

180 = 36·5, so √180 = 6√5, and a = 6.

Order of magnitude — how many times larger is 8 × 10⁹ than 2 × 10⁶?

(8/2) × 10⁹⁻⁶ = 4 × 10³ = 4,000 times.

Fractional power — evaluate 64^(2/3).

(∛64)² = 4² = 16.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

aᵐ · aⁿTap to reveal
aᵐ⁺ⁿ (same base ⇒ add exponents)
(aᵐ)ⁿTap to reveal
aᵐⁿ (multiply exponents)
a⁻ⁿTap to reveal
1/aⁿ (reciprocal; sign unchanged)
a^(m/n)Tap to reveal
ⁿ√(aᵐ) = (ⁿ√a)ᵐ — root first
√(k²·m)Tap to reveal
k√m (pull out perfect squares)
√(a + b)Tap to reveal
≠ √a + √b — roots do not split over a sum
Scientific notation formTap to reveal
a × 10ⁿ with 1 ≤ a < 10
√x for 0 < x < 1Tap to reveal
larger than x (e.g. √0.25 = 0.5)
−2⁴ vs (−2)⁴Tap to reveal
−16 vs 16

📌 Quick revision

Exponents follow five laws that apply only when the bases match, so the master move is rewriting numbers to a common prime base (8 = 2³, 27 = 3³). A negative exponent means reciprocal and a fractional exponent means a root — take the root first to keep the arithmetic calculator-free. Know your perfect squares and cubes, simplify radicals by pulling out square factors, and never split a root over a sum. Watch the GRE’s favourite traps: −2⁴ ≠ (−2)⁴, and for a base between 0 and 1 higher powers shrink while roots grow. Scientific notation (a × 10ⁿ, 1 ≤ a < 10) turns huge or tiny numbers into quick exponent arithmetic and makes order-of-magnitude comparisons immediate.

Chapter test

📝 Exponents & Roots — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara GRE success checklist

You have truly mastered Exponents & Roots when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (5 topics)5/5
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall9 cards