6.2 Exponents, Roots, Estimation, Percent, Ratio & Rate
Key Takeaways
- Five exponent rules: a^m × a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 (a ≠ 0), a^(−n) = 1/a^n — but a^m + a^n does NOT combine.
- Percent change formula: %Δ = (new − old) / old × 100; a +20% rise followed by a −20% fall gives net −4% (1.2 × 0.8 = 0.96), never a return to original.
- Ratio a:b means a/b; with total T and parts a:b summing to s, the parts are (a/s)·T and (b/s)·T.
- Distance = rate × time; combined work rate = sum of individual rates, so time together = 1 / (1/t_A + 1/t_B).
- Estimation tools: front-end addition, compatible numbers (22% of 47 ≈ 20% of 50 = 10), and one-significant-figure rounding.
Exponents, Roots, Estimation, Percent, Ratio & Rate
Quick Answer: Five tools dominate this section: the exponent rules (a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n), percent change = (new − old)/old × 100, ratio a:b means a/b, distance = rate × time, and estimation via compatible numbers. Most GRE word problems reduce to one of these five.
Exponent Rules (Know All Five)
For nonzero a and integers m, n:
| Rule | Form | Example |
|---|---|---|
| Product | a^m × a^n = a^(m+n) | x³ × x⁴ = x⁷ |
| Quotient | a^m ÷ a^n = a^(m−n) | x⁸ / x² = x⁶ |
| Power | (a^m)^n = a^(mn) | (x³)⁵ = x¹⁵ |
| Zero | a^0 = 1 (a ≠ 0) | 17^0 = 1 |
| Negative | a^(−n) = 1 / a^n | 2^(−3) = 1/8 |
Watch for traps: a^m × b^m = (ab)^m, and (a/b)^m = a^m / b^m, but a^m + a^n is NOT a^(m+n) — you cannot combine exponents across addition.
Worked Example — Combine and Simplify
Simplify (x⁵)² × x³ ÷ x⁴.
- (x⁵)² = x¹⁰ (power rule)
- x¹⁰ × x³ = x¹³ (product)
- x¹³ ÷ x⁴ = x⁹ (quotient)
- Answer: x⁹.
Roots and Radicals
For nonnegative a, √a is the nonnegative number whose square is a. Key rules:
- √(ab) = √a × √b (lets you simplify √72 = √(36 × 2) = 6√2)
- √(a/b) = √a / √b
- (√a)² = a
- √a is irrational unless a is a perfect square — a common GRE fact.
Cube roots: ∛a is defined for all real a (negative included): ∛(−8) = −2. Higher even roots (∜a) require nonnegative radicands.
Worked Example — Rationalizing
Simplify 3/√2 to a rationalized denominator.
- Multiply numerator and denominator by √2: (3 × √2) / (√2 × √2) = 3√2 / 2.
- Decimal value ≈ 2.121.
Estimation Strategies
The on-screen calculator does NOT respect order of operations, so estimation often beats punching keys. Three core techniques:
- Front-end: Add leading digits, ignore the rest. 4,318 + 6,702 ≈ 4,000 + 6,000 = 10,000 (true value 11,020).
- Compatible numbers: Pick round numbers close to the originals. 22% of 47 ≈ 20% of 50 = 10 (true 10.34).
- Rounding to one significant figure: 0.49 × 198 ≈ 0.5 × 200 = 100 (true 97.02).
For QC, estimate BOTH quantities first — if one is clearly bigger, pick it without computing exactly. Reserve exact arithmetic for cases where the two estimates overlap.
Percent: Three Formulas, One Idea
Percent means "per 100." Three formulas cover every GRE percent problem:
| Type | Formula |
|---|---|
| Percent of a number | part = (percent / 100) × whole |
| Percent increase | %Δ = (new − old) / old × 100 |
| Percent decrease | %Δ = (old − new) / old × 100 |
Worked Example — Successive Percent Change
A price rises 20%, then falls 20%. Is it back to original? No.
- Let original = 100. After +20%: 120. After −20%: 120 × 0.80 = 96.
- Net change = −4%. Two opposite percent changes never cancel unless both are 0; the multiplier is 1.2 × 0.8 = 0.96.
Worked Example — Percent Greater / Less Than
"X is 30% greater than Y" means X = 1.30Y, or X/Y = 1.30. So Y is what percent of X?
- Y/X = 1/1.30 ≈ 0.769 → Y is about 76.9% of X, so Y is 23.1% less than X. The two percents are NOT the same — a classic GRE trap.
Ratios
A ratio a:b means a/b. With a ratio of 3:5 and total T, the parts are (3/8)T and (5/8)T. Cross-multiply to test equivalence: 3:5 = 6:10 = 12:20.
Worked Example — Ratio Split
A jar has red and blue marbles in ratio 3:5. There are 40 marbles total. How many red?
- 3 + 5 = 8 "parts." Each part = 40/8 = 5.
- Red = 3 × 5 = 15. Blue = 25. (Check 15 + 25 = 40 ✓.)
For three-way ratios (a:b:c), find a common scaling factor. If a:b = 2:3 and b:c = 4:5, rewrite so b matches: a:b = 8:12, b:c = 12:15, so a:b:c = 8:12:15.
Rates: d = rt and Work Problems
Distance = rate × time. Work problems use the same structure: rate = jobs/time.
Worked Example — Combined Work
Pipe A fills a tank in 6 hours; Pipe B fills it in 4 hours. Together, how long?
- Rates: A = 1/6 tank/hr, B = 1/4 tank/hr. Combined = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 tank/hr.
- Time = 1 / (5/12) = 12/5 = 2.4 hours.
Worked Example — Relative Motion
Two cars approach each other at 60 and 40 mph. Closing rate = 60 + 40 = 100 mph. If 250 miles apart, time to meet = 250/100 = 2.5 hours.
Putting It Together: A Combined Estimation + Percent Problem
A shirt originally priced at $80 is marked up 40%, then put on sale for 25% off the new price. Estimate the final price.
- After 40% markup: 80 × 1.4 = 112.
- After 25% off: 112 × 0.75 = 84.
- Final ≈ $84 (net +5% vs original — note: 1.4 × 0.75 = 1.05, so net is +5%, not +15%).
Estimation check: 40% of 80 ≈ 32 (true 32), 25% of 112 ≈ 28 (true 28), 112 − 28 = 84. The estimate matched the exact value, confirming no arithmetic slip.
Simplify (2³ × 2⁵) / 2⁴.
A price increases from $50 to $60. What is the percent increase?
Pipe A fills a tank in 3 hours; Pipe B in 6 hours. Working together, how many hours to fill one tank?
Estimate 0.49 × 198 using compatible numbers. Which value is closest?