AP PRECALCULUS — PREREQUISITE REVIEW
Radicals
Notes — Prerequisite Topic 7
💡 Learning Objectives
By the end of this lesson you will be able to:
- Interpret √x, ∛x, and ⁿ√x as inverse operations of powers
- Simplify radical expressions by extracting perfect-power factors
- Rewrite between radical and rational-exponent form
- Multiply, divide, add, and subtract radical expressions
- Rationalize denominators containing a single radical or a binomial with radicals
- Solve basic radical equations and recognize extraneous solutions
1. What a Radical Means
The n-th root of a, written ⁿ√a, is a number r whose n-th power equals a — that is, rⁿ = a. The number n is the index; when n = 2 it is usually omitted, so √a means the (principal, non-negative) square root of a.
💡 Key definitions
√a = r means r ≥ 0 and r² = a (a ≥ 0 required)
∛a = r means r³ = a (a can be any real number)
ⁿ√a = r means rⁿ = a (for even n, a must be ≥ 0; for odd n, a can be negative)
The graphs of y = √x (defined only for x ≥ 0) and y = ∛x (defined for every real x). Square roots and other even roots have restricted domains; odd roots do not.
⚠️ Even-indexed roots of negatives are not real
√(−4), ⁴√(−16), and so on are not real numbers. They are complex numbers — see the Complex Numbers review.
However, ∛(−8) = −2, because (−2)³ = −8.
2. Radical ↔ Rational Exponent
Every radical is really a rational exponent in disguise. This translation unlocks the exponent rules for work with radicals.
Radical form | Exponent form | Worked example |
|---|---|---|
√x | x^(1/2) | √9 = 9^(1/2) = 3 |
∛x | x^(1/3) | ∛27 = 27^(1/3) = 3 |
ⁿ√x | x^(1/n) | ⁴√16 = 16^(1/4) = 2 |
ⁿ√(xᵐ) | x^(m/n) | ∛(64²) = 64^(2/3) = 16 |
3. Simplifying Radicals
A radical expression is considered simplified when:
- No radicand contains a factor that is a perfect n-th power (for an index-n radical)
- No radicand contains a fraction
- No radical appears in the denominator of a fraction
3.1 Extracting perfect-square factors
Use the product property: √(ab) = √a · √b (for a, b ≥ 0). Break the radicand into a perfect square times something else.
📘 Example — Simplify a square root
Simplify √72.
72 = 36 · 2, and 36 is a perfect square.
√72 = √36 · √2 = 6√2.
📘 Example — Simplify with variables
Simplify √(50x³y⁴), assuming x ≥ 0 and y ≥ 0.
50x³y⁴ = 25 · x² · y⁴ · 2x.
√(50x³y⁴) = 5 · x · y² · √(2x) = 5xy²√(2x).
3.2 Higher-index radicals
For a cube root, look for perfect cubes (1, 8, 27, 64, 125, …). For a fourth root, look for perfect fourth powers (1, 16, 81, 256, …), and so on.
📘 Example — A cube root
Simplify ∛(54x⁴).
54 = 27 · 2 and x⁴ = x³ · x, so ∛(54x⁴) = ∛(27x³) · ∛(2x) = 3x · ∛(2x).
4. Arithmetic with Radicals
4.1 Adding and subtracting
Only like radicals — radicals with the same index and the same radicand — can be combined. Treat them like variables when grouping.
📘 Example — Combine like radicals
Simplify 3√2 + 5√2 − √2 = (3 + 5 − 1)√2 = 7√2.
Simplify √12 + √27. First simplify each:
√12 = 2√3, √27 = 3√3, so the sum is 5√3.
4.2 Multiplying and dividing
Use √a · √b = √(ab) and √a / √b = √(a/b) (for b > 0). Distribute as with polynomials when multiple terms appear.
📘 Example — Multiplying radicals
(3 + √5)(2 − √5) = 3 · 2 − 3√5 + 2√5 − (√5)² = 6 − √5 − 5 = 1 − √5.
5. Rationalizing Denominators
Convention says that no denominator should contain a radical. Two standard techniques handle this.
5.1 Monomial denominator
Multiply the numerator and denominator by whatever is needed to turn the denominator into a rational number.
📘 Example — Single radical in the denominator
Simplify 5 / √7 = (5 / √7) · (√7 / √7) = 5√7 / 7.
Simplify 3 / ∛2. Multiply by ∛4 / ∛4: 3 · ∛4 / ∛8 = 3∛4 / 2.
5.2 Binomial denominator — use the conjugate
The conjugate of a + b is a − b. Multiplying by the conjugate uses the difference-of-squares identity to eliminate the radical.
📘 Example — Rationalizing with a conjugate
Simplify 4 / (3 + √5).
Multiply top and bottom by (3 − √5):
= 4(3 − √5) / ((3 + √5)(3 − √5)) = 4(3 − √5) / (9 − 5) = (12 − 4√5)/4 = 3 − √5.
6. Solving Radical Equations
To solve an equation in which the variable appears under a radical:
- Step 1: isolate the radical.
- Step 2: raise both sides to the power matching the index.
- Step 3: solve the resulting equation.
- Step 4: check every candidate in the original equation — some answers may be extraneous.
⚠️ Why check for extraneous solutions?
Squaring both sides of an equation can introduce solutions that do not work in the original, because squaring erases sign information. Always verify.
📘 Example — A radical equation
Solve √(2x + 3) = x.
Square: 2x + 3 = x² ⟹ x² − 2x − 3 = 0 ⟹ (x − 3)(x + 1) = 0.
Candidates x = 3 or x = −1.
Check x = 3: √9 = 3 ✓. Check x = −1: √1 = 1, but the right-hand side is −1. ✗.
The only solution is x = 3.
7. Summary
- Radicals are rational exponents; translate freely between the two forms
- Simplify by removing perfect n-th power factors from under the radical
- Combine like radicals as you would like terms
- Rationalize denominators with a clever multiplication (conjugate for binomials)
- When solving, isolate the radical, raise to a power, and always check for extraneous solutions