AP PRECALCULUS โ PREREQUISITE REVIEW
Radicals
Notes โ Prerequisite Topic 7
๐ก Learning Objectives
By the end of this lesson you will be able to:
- Interpret โx, โx, and โฟโx 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 โฟโa, is a number r whose n-th power equals a โ that is, rโฟ = a. The number n is the index; when n = 2 it is usually omitted, so โa means the (principal, non-negative) square root of a.
๐ก Key definitions
โa = r means r โฅ 0 and rยฒ = a (a โฅ 0 required)
โa = r means rยณ = a (a can be any real number)
โฟโa = r means rโฟ = a (for even n, a must be โฅ 0; for odd n, a can be negative)
The graphs of y = โx (defined only for x โฅ 0) and y = โx (defined for every real x). Square roots and other even roots have restricted domains; odd roots do not.
โ ๏ธ Even-indexed roots of negatives are not real
โ(โ4), โดโ(โ16), and so on are not real numbers. They are complex numbers โ see the Complex Numbers review.
However, โ(โ8) = โ2, because (โ2)ยณ = โ8.
2. Radical โ 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 |
|---|---|---|
โx | x^(1/2) | โ9 = 9^(1/2) = 3 |
โx | x^(1/3) | โ27 = 27^(1/3) = 3 |
โฟโx | x^(1/n) | โดโ16 = 16^(1/4) = 2 |
โฟโ(xแต) | x^(m/n) | โ(64ยฒ) = 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: โ(ab) = โa ยท โb (for a, b โฅ 0). Break the radicand into a perfect square times something else.
๐ Example โ Simplify a square root
Simplify โ72.
72 = 36 ยท 2, and 36 is a perfect square.
โ72 = โ36 ยท โ2 = 6โ2.
๐ Example โ Simplify with variables
Simplify โ(50xยณyโด), assuming x โฅ 0 and y โฅ 0.
50xยณyโด = 25 ยท xยฒ ยท yโด ยท 2x.
โ(50xยณyโด) = 5 ยท x ยท yยฒ ยท โ(2x) = 5xyยฒโ(2x).
3.2 Higher-index radicals
For a cube root, look for perfect cubes (1, 8, 27, 64, 125, โฆ). For a fourth root, look for perfect fourth powers (1, 16, 81, 256, โฆ), and so on.
๐ Example โ A cube root
Simplify โ(54xโด).
54 = 27 ยท 2 and xโด = xยณ ยท x, so โ(54xโด) = โ(27xยณ) ยท โ(2x) = 3x ยท โ(2x).
4. Arithmetic with Radicals
4.1 Adding and subtracting
Only like radicals โ radicals with the same index and the same radicand โ can be combined. Treat them like variables when grouping.
๐ Example โ Combine like radicals
Simplify 3โ2 + 5โ2 โ โ2 = (3 + 5 โ 1)โ2 = 7โ2.
Simplify โ12 + โ27. First simplify each:
โ12 = 2โ3, โ27 = 3โ3, so the sum is 5โ3.
4.2 Multiplying and dividing
Use โa ยท โb = โ(ab) and โa / โb = โ(a/b) (for b > 0). Distribute as with polynomials when multiple terms appear.
๐ Example โ Multiplying radicals
(3 + โ5)(2 โ โ5) = 3 ยท 2 โ 3โ5 + 2โ5 โ (โ5)ยฒ = 6 โ โ5 โ 5 = 1 โ โ5.
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.
๐ Example โ Single radical in the denominator
Simplify 5 / โ7 = (5 / โ7) ยท (โ7 / โ7) = 5โ7 / 7.
Simplify 3 / โ2. Multiply by โ4 / โ4: 3 ยท โ4 / โ8 = 3โ4 / 2.
5.2 Binomial denominator โ use the conjugate
The conjugate of a + b is a โ b. Multiplying by the conjugate uses the difference-of-squares identity to eliminate the radical.
๐ Example โ Rationalizing with a conjugate
Simplify 4 / (3 + โ5).
Multiply top and bottom by (3 โ โ5):
= 4(3 โ โ5) / ((3 + โ5)(3 โ โ5)) = 4(3 โ โ5) / (9 โ 5) = (12 โ 4โ5)/4 = 3 โ โ5.
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 โ some answers may be extraneous.
โ ๏ธ 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.
๐ Example โ A radical equation
Solve โ(2x + 3) = x.
Square: 2x + 3 = xยฒ โน xยฒ โ 2x โ 3 = 0 โน (x โ 3)(x + 1) = 0.
Candidates x = 3 or x = โ1.
Check x = 3: โ9 = 3 โ. Check x = โ1: โ1 = 1, but the right-hand side is โ1. โ.
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