Square roots and nth roots

A root is the inverse of raising to a power: since 3² = 9, √9 = 3. Roots appear constantly in geometry (Pythagoras, diagonals, distances), in physics (speeds, periods of oscillation) and in solving equations. Exam questions regularly ask you to evaluate roots, take factors out of a root or bring them in, rationalise denominators, compare roots and simplify “nested roots”. This lesson explains the reason for each rule and the most common traps.

§1. The square root

√a (a ≥ 0) is the non-negative number whose square is a. This is the principal root: √25 = 5, not −5! The equation x² = 25 has two roots (x = ±5), but the symbol √25 means only the positive one.

√(a²) = |a| is the most often forgotten property. Reason: even for negative a, a² is positive, and the root returns a non-negative number. For example, √((−3)²) = √9 = 3, √((1 − √2)²) = |1 − √2| = √2 − 1, since 1 − √2 < 0.
A negative number has no real square root: no real number squared is negative.

§2. The n-th root

ⁿ√a is the number whose n-th power is a.

• For even n: we need a ≥ 0, and the result is ≥ 0: ⁴√16 = 2.
• For odd n: any a, and the sign is kept: ∛(−27) = −3, because (−3)³ = −27.

Switching to powers: ⁿ√(aᵐ) = a^(m/n) (a > 0). This lets you handle roots with the laws of indices: √a·∛a = a^(1/2 + 1/3) = a^(5/6) = ⁶√(a⁵).

§3. Properties and transformations

• Taking a factor out: write the number under the root as “perfect square × remainder”: √72 = √(36·2) = 6√2.
• Bringing a factor in: the factor is squared: 3√2 = √(9·2) = √18.
• Like surds (same number under the root) are added like like terms: √12 + √27 − √48 = 2√3 + 3√3 − 4√3 = √3.

Careful: √(a + b) ≠ √a + √b. Check: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.

Worked example

Simplify √50 − √18 + √8.

1. √50 = √(25·2) = 5√2.
2. √18 = √(9·2) = 3√2, √8 = √(4·2) = 2√2.
3. 5√2 − 3√2 + 2√2 = 4√2.

§4. Rationalising the denominator

If a denominator contains a root, it is customary to rewrite the fraction without a root below — fractions are then easier to compare and add.

• Denominator √a: multiply top and bottom by √a: 6/√3 = 6√3/3 = 2√3.
• Denominator ∛a: multiply by ∛(a²): 1/∛2 = ∛4/2.
• Denominator √a ± b or √a ± √b: multiply by the conjugate, because (√a − b)(√a + b) = a − b².

Worked example

Simplify (√5 + √3)/(√5 − √3).

1. Multiply top and bottom by √5 + √3.
2. Denominator: (√5)² − (√3)² = 5 − 3 = 2.
3. Numerator: (√5 + √3)² = 5 + 2√15 + 3 = 8 + 2√15.
4. Result: (8 + 2√15)/2 = 4 + √15.

§5. Nested roots

Idea: write the expression under the root as a perfect square, since (√x ± √y)² = x + y ± 2√(xy). First make the coefficient in front of the inner root equal to 2: 7 − 4√3 = 7 − 2√12; x + y = 7, xy = 12 → 4 and 3: √(7 − 4√3) = √4 − √3 = 2 − √3.

Worked example

Simplify √(11 + 6√2).

1. 6√2 = 2√18, so x + y = 11, xy = 18.
2. The numbers are 9 and 2.
3. √(11 + 6√2) = √9 + √2 = 3 + √2.

Worked example (from the official sample)

Calculate (√(4 − √7) + √(4 + √7))².

1. (A + B)² = A² + B² + 2AB.
2. A² + B² = (4 − √7) + (4 + √7) = 8.
3. AB = √((4 − √7)(4 + √7)) = √(16 − 7) = 3.
4. Result: 8 + 2·3 = 14.

§6. Comparing roots

Roots of the same degree are compared by the numbers under them: √7 < √8. Roots of different degrees are brought to a common degree, or both are raised to the power equal to the LCM of the degrees.

Worked example

Compare √3 and ∛5.

1. Raise both to the 6th power: (√3)⁶ = 3³ = 27.
2. (∛5)⁶ = 5² = 25.
3. 27 > 25, so √3 > ∛5.

§7. Simplifying algebraic expressions with roots

In expressions with letters it helps to treat a root as a “new variable”: a = (√a)² (a ≥ 0). Then the special product formulas apply:

This allows fractions to be simplified, but you must remember the domain (a ≥ 0, denominator ≠ 0).

Worked example

Simplify (a − 4)/(√a + 2) (a ≥ 0).

1. Write the numerator as a difference of squares: a − 4 = (√a)² − 2² = (√a − 2)(√a + 2).
2. √a + 2 cancels (it is never zero).
3. Answer: √a − 2.

Worked example

Calculate (√3 + 1)² − √12.

1. (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3.
2. √12 = 2√3.
3. 4 + 2√3 − 2√3 = 4.

Check yourself

1. Calculate √12 + √27 − √48.

2. Simplify 6/√3.

3. Simplify 1/(√5 − 2).

4. Simplify √(7 − 4√3).

5. Calculate ∛(−27) + ⁴√81.

6. Simplify √((1 − √2)²).

7. Which is larger: 2√7 or 3√3?

8. Simplify √(x² − 6x + 9) for x < 3.

Common mistakes

• Writing √25 = ±5 — the principal root is non-negative.
• Writing √(x²) = x — it is |x|.
• Assuming √(a + b) = √a + √b.
• Bringing a factor under the root without squaring it: 3√2 ≠ √6.
• Trying to take an even root of a negative number: √(−4) does not exist among the reals.
• Multiplying only the denominator by the conjugate and forgetting the numerator.