Surds and simplification

Learning outcome

Simplify square-root surds, combine and multiply radical expressions, and rationalise denominators while respecting real-number domains.

Concepts and assumptions

Work over the real numbers. The square-root symbol denotes the principal, nonnegative square root. Thus √a requires a ≥ 0. Here, square-root surds are irrational square roots of rational numbers: √2 is a surd, but √9 = 3 is not.

For a, b ≥ 0, √a × √b = √(ab). Both sides are nonnegative and their squares equal ab, which explains the rule. Similarly, √(a/b) = √a/√b requires a ≥ 0 and b > 0; the denominator cannot vanish.

Extract perfect-square factors before combining terms. Only like surds combine: their irrational parts must match after simplification. Distribution explains this just as it explains combining like algebraic terms.

Square roots do not distribute over addition. Even for nonnegative a and b, √(a + b) generally differs from √a + √b.

For every real x, √(x²) = |x|, not always x. Here |x| equals x when x ≥ 0 and −x when x < 0. The nonnegative-root convention forces this distinction.

Rationalising rewrites a fraction with no surd in its denominator. Multiplying numerator and denominator by the same nonzero expression multiplies the fraction by 1. Conjugates use opposite signs: (u + v)(u − v) = u² − v². Check that both the original denominator and the proposed multiplying factor are nonzero.

Keep surds exact unless a decimal approximation is requested; no rounding is needed here.

Worked examples

Example 1 — Combining simplified surds. Simplify √72 + √50 − √8.

Extract square factors: √72 = √(36 × 2) = 6√2; √50 = √(25 × 2) = 5√2; √8 = √(4 × 2) = 2√2. Therefore the expression equals (6 + 5 − 2)√2 = 9√2.

Example 2 — Squaring a surd sum. Simplify (√6 + √2)².

Use distribution: (√6)² + 2√6√2 + (√2)² = 6 + 2√12 + 2. Since √12 = 2√3, the result is 8 + 4√3. The cross term cannot be omitted.

Example 3 — Rationalising a difference. Simplify 4/(√7 − √3).

The denominator is positive because √7 > √3; the conjugate √7 + √3 is also positive. Multiply by (√7 + √3)/(√7 + √3): 4(√7 + √3)/(7 − 3) = 4(√7 + √3)/4 = √7 + √3.

Common mistakes

Adding radicands instead of like surds; forgetting cross terms; treating √(x²) as x for negative x; using a zero rationalising factor; introducing unnecessary rounded decimals.

Practice questions

1. Simplify √108 − √48 + √12.
2. Expand and simplify (√5 + √2)².
3. Rationalise 3/(√5 + √2).
4. For real x, state the domain and simplify √(x²)/x.

Worked solutions

1. √108 = 6√3, √48 = 4√3 and √12 = 2√3. Thus (6 − 4 + 2)√3 = 4√3.
2. Expansion gives 5 + 2√10 + 2 = 7 + 2√10. Both radicands are nonnegative.
3. The denominator is positive, and √5 − √2 is nonzero. Multiply by the conjugate ratio: 3(√5 − √2)/(5 − 2) = √5 − √2.
4. The radicand x² is nonnegative for every real x, but division excludes x = 0. The expression is |x|/x: it equals 1 when x > 0, and −1 when x < 0, because then |x| = −x.