Exponents & Radicals
Exponents tell you how many times to multiply a base by itself. Radicals are the inverse — finding what number was repeatedly multiplied.
Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x² = x⁵ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³ |
| Power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero | a⁰ = 1 | 7⁰ = 1 |
| Negative | a⁻ⁿ = 1/aⁿ | x⁻² = 1/x² |
Radicals
√n means "what squared gives n?" Cube root ∛n means "what cubed gives n?"
√49 = 7 | ∛27 = 3 | √72 = √(36×2) = 6√2
Rational Exponents
a^(m/n) = (ⁿ√a)ᵐ
8^(2/3) = (∛8)² = 2² = 4
25^(1/2) = √25 = 5
Simplifying Radical Expressions
Factor out perfect squares (or cubes). Combine like radicals just like like terms.
3√2 + 5√2 = 8√2 | √50 = 5√2
Why the Rules Work
Every exponent rule comes from writing the powers out the long way. You never have to memorize a rule you can rebuild in ten seconds.
Where the product rule comes from
x³ × x² means (x·x·x) × (x·x), which is five x's multiplied together — so x⁵. You added the exponents because you were counting how many x's ended up in the pile.
Where the zero rule comes from
x³ ÷ x³ is a number divided by itself, which is 1. The quotient rule says it is x³⁻³ = x⁰. Both are correct, so x⁰ must equal 1.
Step by Step: Simplifying a Radical
To simplify a square root, pull out the largest perfect square hiding inside it.
Simplify √180
- Find the largest perfect square that divides 180. Try 36:
180 ÷ 36 = 5. - Split the radical:
√180 = √(36 × 5) = √36 × √5 - Take the root you know:
√36 = 6 - Answer:
6√5
If you had used 4 instead of 36, you would get 2√45 — not wrong, just not finished. Keep factoring until nothing square is left inside.
Common Mistakes
√(a + b)is not√a + √b. Check it:√(9+16) = √25 = 5, but3 + 4 = 7.- A negative exponent does not make the answer negative.
2⁻³ = 1/8, a positive number. −3²and(−3)²differ. Without parentheses the exponent applies only to the 3, giving−9. With them you get9.- Only like radicals combine.
3√2 + 5√2 = 8√2, but√2 + √3stays as it is.
Where You Will Use This
- The Pythagorean theorem gives answers like
√50, which you simplify to5√2. - Compound interest and population growth are exponent problems.
- Scientific notation is exponent rules applied to powers of 10.
- Distance, area, and standard deviation formulas all end in square roots.
Summary
- Multiplying like bases adds exponents; dividing subtracts them.
- Anything to the zero power is 1; a negative exponent means "one over".
- A radical is the inverse of a power, and
a^(m/n)is just both at once. - Simplify a radical by factoring out the biggest perfect square.
- Radicals add only when the part under the root already matches.
FAQ
Can you add √2 + √3? No — they are unlike radicals. Leave as is.
What is (−2)⁴? 16 — even exponent makes it positive.
What does a fractional exponent like 1/2 actually mean? It means a root. a^(1/2) is the square root of a, and a^(1/3) is the cube root. The bottom number of the fraction tells you which root; the top number tells you which power.
Can you take the square root of a negative number? Not within the real numbers, because no real number squared is negative. In advanced classes these become imaginary numbers, written with i.
Why is anything to the zero power 1 and not 0? Because dividing a power by itself gives 1, and the quotient rule turns that same division into a zero exponent. Both descriptions have to agree.
How do I know when a radical is fully simplified? When no perfect square (or perfect cube, for cube roots) divides the number left under the radical sign.
Quick Quiz
Test what you just learned. Choose the best answer for each question.