What is an nth root?
A root undoes a power. If b × b = a, then b is a square root of a. If b × b × b = a, then b is the cube root of a.
In general, ⁿ√a is the number that gives a when it is multiplied by itself n times. The sign √ is called the radical sign, a is the radicand and n is the index.
- √49 = 7 because 7² = 49.
- ∛125 = 5 because 5³ = 125.
- ⁴√81 = 3 because 3⁴ = 81.
Even and odd index
For an even index (2, 4, 6…), the radicand must not be negative if we want a real answer: √(−9) is not a real number. The symbol √ always means the non-negative root, so √9 = 3 (not −3), even though (−3)² = 9 too.
For an odd index (3, 5…), negative numbers are fine: ∛(−8) = −2.
Also, √(x²) = |x|, not x. For example √((−5)²) = √25 = 5.
Rules of radicals and simplifying
For a, b ≥ 0:
- Product rule: ⁿ√a × ⁿ√b = ⁿ√(ab)
- Quotient rule: ⁿ√a ÷ ⁿ√b = ⁿ√(a/b), b ≠ 0
- Power rule: (ⁿ√a)^m = ⁿ√(a^m)
- Root of a root: ᵐ√(ⁿ√a) = ᵐⁿ√a
Simplifying
Pull out the biggest perfect power. √72 = √(36 × 2) = 6√2. ∛54 = ∛(27 × 2) = 3∛2.
Adding and subtracting
Only like radicals (same index, same radicand) can be added: 3√2 + 5√2 = 8√2. But √2 + √3 cannot be joined. Simplify first: √50 + √8 = 5√2 + 2√2 = 7√2.
Warning: √(a + b) is NOT √a + √b. Check: √(9 + 16) = 5, but 3 + 4 = 7.
Rationalising the denominator
We usually do not leave a root in the denominator. To remove it, multiply top and bottom by the same thing (this is multiplying by 1, so the value does not change).
- Single root: 3/√5 = 3√5/5.
- Cube root: 1/∛2 = ∛4/∛8 = ∛4/2.
- Two terms: use the conjugate. For 1/(√3 − 1), multiply by (√3 + 1): (√3 + 1)/(3 − 1) = (√3 + 1)/2. This works because (a − b)(a + b) = a² − b².
Rational exponents: roots as powers
A root can be written as a power with a fraction:
a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)
Example: 32^(3/5) = (⁵√32)³ = 2³ = 8. And 27^(−2/3) = 1/(∛27)² = 1/9.
All the laws of exponents still work: a^p × a^q = a^(p+q), (a^p)^q = a^(pq). This makes many radical problems easy: √2 × ∛2 = 2^(1/2 + 1/3) = 2^(5/6) = ⁶√32.
Radical function
The function y = ⁿ√x is the opposite (inverse) of y = xⁿ. For even n it lives only for x ≥ 0 and rises slowly; for odd n it works for all x.
Try it: roots with real objects
Take 36 coins or buttons. Can you make a square? What is the side? Now try 40: you get a 6 × 6 square and 4 left over, so √40 is a bit more than 6 (about 6.32). With small cubes (or sugar cubes) try 8, 27 and 30. In the 3D free play, check your guesses.
Key formulas and definitions
- ⁿ√a = a^(1/n)
- a^(m/n) = (ⁿ√a)^m
- ⁿ√a × ⁿ√b = ⁿ√(ab)
- ⁿ√a ÷ ⁿ√b = ⁿ√(a/b)
- √(x²) = |x|
- 1/(√a − √b) = (√a + √b)/(a − b)
Worked examples
1. Find ⁴√625.
5⁴ = 625, so ⁴√625 = 5.
2. Simplify √98.
98 = 49 × 2, so √98 = √49 × √2 = 7√2.
3. Simplify ∛250.
250 = 125 × 2, so ∛250 = 5∛2.
4. Simplify 2√12 + √27.
2√12 = 2 × 2√3 = 4√3; √27 = 3√3. Total = 7√3.
5. Rationalise 6/√3.
6/√3 × √3/√3 = 6√3/3 = 2√3.
6. Rationalise 4/(√5 + 1).
Multiply by (√5 − 1): 4(√5 − 1)/(5 − 1) = √5 − 1.
7. Evaluate 16^(3/4).
⁴√16 = 2, then 2³ = 8.
8. Write √x × ∛x as one power of x.
x^(1/2) × x^(1/3) = x^(5/6) = ⁶√(x⁵).
Common mistakes
- Writing √(a + b) = √a + √b. Roots split over × and ÷ only, never over + or −.
- Saying √16 = ±4. The √ sign means the non-negative root, so √16 = 4. The equation x² = 16 has two answers, ±4.
- Adding unlike radicals: √2 + √3 is not √5.
- Mixing up a^(m/n): the bottom number n is the root, the top number m is the power.