Surd Survival

Surd Survival

On Paper 1 (non-calculator) a root like √2 has no tidy decimal, so you leave it as a surd and work with it exactly. This is the survival kit: simplify, multiply, add, rationalise, and crack awkward indices, all by hand.

Spot the surd

Which of these is a surd (an irrational root)?

• √20
• √36
• √64
• √81

Simplifying surds

To simplify a surd, split it using its largest square factor, then take the root of that part: √12 = √(4 × 3) = √4 × √3 = 2√3. The trick is spotting the biggest square that divides in: - √50 = √(25 × 2) = 5√2 - √72 = √(36 × 2) = 6√2 Pick the *largest* square factor, or you will have to simplify twice. √72 = √(4 × 18) = 2√18 is not wrong, but it is not finished either.

Simplify √48

Simplify √48 into the form k√3. What is the value of k?

Multiplying surds

Roots multiply straight across: √a × √b = √(ab), and often the result is a whole number. √3 × √12 = √36 = 6. ⚠️ But roots do not split over addition: √(a + b) is NOT √a + √b. √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. Five is not seven, so never break a root across a plus sign. The same multiplying rule lets you rationalise a denominator. A surd on the bottom is untidy, so multiply top and bottom by that surd, which does not change the value: 1/√2 = √2 / (√2 × √2) = √2 / 2. It often tidies further: 6/√3 = 6√3 / 3 = 2√3.

Multiply them

Using √a × √b = √(ab): √2 × √8 = √(_____ × _____) = √_____ = _____. The two surds have multiplied into a whole number, which is exactly what makes this rule worth reaching for.

Mind the trap

What is √(36 + 64)?

• 10
• 14
• 100
• 50

Rationalise 10/√5

Rationalising 10/√5 gives the form k√5. What is the value of k?

Adding surds

Surds add and subtract exactly like algebra. The only question is whether the roots match, and the answer decides whether anything can be done at all:

Simplify, then add

Work out √27 + √12, giving your answer in the form k√3. Simplify each surd first. What is k?

Surd survival rules

To simplify a surd, take out the largest _____ factor. You can only add or subtract _____ surds, so 3√2 + 5√2 = 8√2 but √2 + √3 will not combine. And remember √(a + b) is _____ equal to √a + √b.

Fractional and negative indices

Indices follow fixed laws. Two of them trip people up: - A negative index means reciprocal: a⁻ⁿ = 1/aⁿ. - A fractional index means a root: the bottom of the fraction is the root and the top is the power, so a^(m/n) = (ⁿ√a)ᵐ. Chain them for the classic grade-9 evaluation: 27^(−2/3) = 1 / 27^(2/3) = 1 / (³√27)² = 1 / 3² = 1/9. Do the root BEFORE the power and the numbers stay small. Doing it the other way round leaves you cube-rooting 729 by hand for no reason.

Order the working

Put the steps to evaluate 81^(−3/4) in the right order.

• Rewrite the negative index as a reciprocal: 81^(−3/4) = 1 ÷ 81^(3/4)
• Use the bottom of the power as a root: ⁴√81 = 3
• Apply the top of the power: 3³ = 27
• Combine: 81^(3/4) = 27, so the answer is 1/27

Match each power to its value

Show your working

Evaluate 64^(−2/3) without a calculator, showing every step of your working. Give your answer as a fraction.

• Deal with the negative index first, and say what it becomes
• Say which part of the fraction 2/3 is the root and which is the power
• Show the value of the root before you apply the power, so the numbers stay small
• Give the final answer as a fraction in its simplest form