Revision notes
Keep square roots exact: pull out square factors, combine like surds and clear roots from the bottom of a fraction.
Emeka is tiling a square floor with area 2 m². Its side is √2 m — a decimal that never ends or repeats. Instead of rounding, mathematicians keep it exact as √2. Numbers like this are called surds.
A surd is a root that cannot be written as an exact whole number or fraction, like √2, √3 or √5.
√9 = 3 and √25 = 5 are not surds — they are whole numbers.
We leave surds as roots so answers stay exact; rounding 1.414… would add a small error.
🔑 Key idea: A surd is an exact root that is not a whole number or fraction.
Use the rule √(a × b) = √a × √b. Split the number into a square number × another number.
√72 = √(36 × 2) = √36 × √2 = 6√2.
Always use the largest square factor. √72 = √(4 × 18) = 2√18 is not finished, because √18 = 3√2 can still be simplified.
🔑 Key idea: √(a × b) = √a × √b — take out the largest square factor.
Like surds have the same number under the root. Add them like objects: 3√5 + 2√5 = 5√5. But √2 + √3 is not √5.
To multiply, multiply outside by outside and inside by inside: 2√3 × 5√2 = 10√6. Remember √3 × √3 = 3.
To rationalise 6 ÷ √3, multiply top and bottom by √3: 6√3 ÷ 3 = 2√3. The value doesn't change, but the root leaves the bottom.
🔑 Key idea: Only like surds can be added. √a × √a = a — use this to clear a root from the denominator.
Simplify √75.
Step 1: Split into a square number × another number
√75 = √(25 × 3)
Step 2: Take the square root of the square
= √25 × √3 = 5√3
Answer: 5√3
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