Surds and Radicals: Simplifying √ Expressions with Examples

A surd is a root of a number that cannot be simplified into a rational number, such as the square root of 2 or the cube root of 5. The symbol √ is called the radical sign, and the number inside it is the radicand. Surds appear whenever a perfect square, cube or higher power is missing — √2 is the exact length of the diagonal of a 1 x 1 square, and no fraction can express it.

Surds matter because exact answers beat decimal approximations in mathematics. Writing the diagonal as √2 keeps it perfectly exact, while 1.4142 is only an approximation that accumulates error through a long calculation. Geometry, trigonometry and quadratic equations constantly produce surds, so handling them cleanly is a core skill.

Key Concepts

• Radical sign (√): denotes the principal (positive) square root. A small 3 in front denotes cube root.
• Radicand: the number under the radical sign.
• Perfect square: numbers like 4, 9, 16 and 25 whose square roots are whole numbers.
• Simplest surd form: a surd with no perfect-square factor left inside, for example 2√3 instead of √12.
• Like surds: surds with the same irrational part, like 3√2 and 5√2, which can be added like like terms.
• Conjugate: for a + √b, the conjugate is a − √b; their product is rational (used in rationalising denominators).

Formulas

• √a x √b = √(ab), and √a ÷ √b = √(a/b), for a, b ≥ 0.
• (√a)² = a.
• Simplifying: split the radicand into a perfect square times the remainder, for example √12 = √(4 x 3) = 2√3.
• Addition of like surds: p√a + q√a = (p + q)√a.

Worked Example

Simplify √50 − √18.

1. Split each into perfect squares: √50 = √(25 x 2) = 5√2, and √18 = √(9 x 2) = 3√2.
2. Both are now like surds with irrational part √2, so subtract the coefficients: 5√2 − 3√2 = (5 − 3)√2 = 2√2.
3. Check numerically: √50 ≈ 7.071, √18 ≈ 4.243, difference ≈ 2.828, and 2√2 ≈ 2 x 1.414 = 2.828. Correct.

Common Misconceptions

• “√a + √b = √(a + b).” Wrong. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.
• “Surds are just rounded decimals.” Wrong. √2 is an exact value; 1.414 is the approximation. Replacing surds early throws away exactness.
• “√(−4) = −2.” Wrong. No real number squares to a negative, so the square root of a negative is not a real number at all.

Real-World Applications

• Geometry: diagonals, the height of an equilateral triangle (h = (√3/2) x side), and the Pythagoras theorem produce surds constantly.
• Electrical engineering: AC circuit calculations use √2 (RMS values) everywhere.
• Exact trigonometry: sin 60° = √3/2 and cos 45° = 1/√2 are surd values used in navigation and construction.

A surd is an exact root that refuses to become a neat fraction. Pull out the perfect-square factors, combine only like surds, and never split a root over addition. Keep answers in surd form until the final step, and your results stay perfectly exact.