Exponents & Roots • Topic 1 of 5

Laws of Exponents

Five rules do almost all the work. Multiply same-base powers by adding exponents (a³·a⁴ = a⁷); divide by subtracting (a⁷/a⁴ = a³); raise a power to a power by multiplying ((a³)⁴ = a¹²); distribute a power over a product ((2x)³ = 8x³); and remember a⁰ = 1 for any non-zero a. The catch: these apply only when the BASES match, so a²·b³ does not simplify. The GRE’s favourite trick is to hide a common base — 8 is 2³, 27 is 3³, 25 is 5² — so rewriting everything to a common prime base is often the whole solution. When an equation has the same base on both sides, you can simply set the exponents equal: if 2ˣ = 2⁵ then x = 5. Keep a sharp eye on order of operations too: −2⁴ means −(2⁴) = −16, while (−2)⁴ = 16.

✅ Solved examples

1. Simplify 3⁴ · 3⁵.
Same base, add exponents: 3⁴⁺⁵ = 3⁹.
2. Simplify (2x³)⁴.
Distribute the power: 2⁴ · x¹² = 16x¹².
3. Solve for x: 2ˣ = 32.
32 = 2⁵, so 2ˣ = 2⁵ ⇒ x = 5.
4. Simplify (2¹⁰ · 6⁵) / (4⁴ · 3⁵).
Write in primes: 6⁵ = 2⁵·3⁵, 4⁴ = 2⁸. Numerator = 2¹⁵·3⁵, denominator = 2⁸·3⁵. Ratio = 2¹⁵⁻⁸ = 2⁷ = 128.

✏️ Practice — try these, take hints as needed

1. Simplify 5⁷ / 5³.
Same base, subtract exponents.
5⁷⁻³.
5⁴.
5⁴ = 625
2. Simplify (a⁴)³ · a².
Power of a power first.
(a⁴)³ = a¹².
Then add: a¹² · a².
a¹⁴
3. Solve for x: 3ˣ⁺¹ = 81.
81 = 3⁴.
Set exponents equal.
x + 1 = 4.
x = 3
4. Evaluate −3⁴ and (−3)⁴.
−3⁴ means −(3⁴).
3⁴ = 81.
Even power of a negative is positive.
−81 and 81
5. Simplify (4³ · 8²) / 2⁵.
Convert to base 2.
4³ = 2⁶, 8² = 2⁶.
2¹² / 2⁵.
2⁷ = 128

📝 Topic test — 8 questions

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