Exponents & Roots • Topic 3 of 5

Square & Cube Roots

A square root asks "what non-negative number squared gives this?" — by convention √25 = 5, not ±5 (the symbol always denotes the principal, non-negative root on the GRE). A cube root asks the same for cubing and, unlike square roots, is defined for negatives: ∛(−8) = −2. It pays to know the perfect squares to about 15² = 225 and the small perfect cubes (1, 8, 27, 64, 125) cold, because the on-screen calculator gives clumsy decimals and the GRE builds its numbers around these. Two facts settle many Quantitative Comparisons: for a number greater than 1, its square root is smaller than itself (√9 = 3 < 9), but for a number between 0 and 1 the square root is LARGER (√0.25 = 0.5 > 0.25). To estimate a messy root, trap it between the two nearest perfect squares: √50 is between 7 and 8, and close to 7.1 since 7.1² ≈ 50.4.

✅ Solved examples

1. Evaluate √196.
14² = 196, so √196 = 14.
2. Evaluate ∛(−64).
Cube roots keep the sign: (−4)³ = −64, so ∛(−64) = −4.
3. Between which two consecutive integers does √72 lie, and which is it nearer?
8² = 64 and 9² = 81, so √72 is between 8 and 9. Test the midpoint: 8.5² = 72.25, just above 72, so √72 ≈ 8.49 — nearer to 8.
4. Quantitative Comparison. Quantity A: √0.16. Quantity B: 0.16. Which is greater?
√0.16 = 0.4. Since 0.4 > 0.16, Quantity A is greater — the "root of a number below 1 is larger" rule.

✏️ Practice — try these, take hints as needed

1. Evaluate √225.
Look for the perfect square.
15² = 225.
Principal root.
15
2. Evaluate ∛125.
What cubed gives 125?
5³ = 125.
Take the cube root.
5
3. Evaluate ∛(−27).
Cube roots allow negatives.
(−3)³ = −27.
Keep the sign.
−3
4. Between which two integers does √130 lie?
Find nearby perfect squares.
11² = 121, 12² = 144.
130 is between them.
between 11 and 12
5. Compare √0.09 and 0.09 — which is larger?
√0.09 = 0.3.
Root of a number below 1 grows.
0.3 vs 0.09.
√0.09 = 0.3 is larger

📝 Topic test — 8 questions

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