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*. Grade 9s handle surds and awkward **indices** fluently, showing each rule. This is your survival kit: simplify surds, multiply and add them safely, rationalise denominators, and crack fractional and negative powers — all by hand.

What is a surd 🌱

A **surd** is a root whose value is **irrational** — it cannot be written as an exact fraction or a terminating decimal. So √2, √3 and √5 are surds; you leave them as they are. But √9 = 3 and √25 = 5 are **not** surds — they simplify to whole numbers. Always check whether a root is a **perfect square** before treating it as a surd.

Spot the surd

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

• √7
• √16
• √25
• √49

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 (4, 9, 16, 25, 36 ...): • √50 = √(25 × 2) = **5√2** • √72 = √(36 × 2) = **6√2** Pick the *largest* square factor or you will have to simplify twice.

Simplify √50

An interactive activity.

Multiplying surds ✖️

Roots multiply straight across: **√a × √b = √(ab)**. Often the result is a whole number: **√3 × √12** = √(3 × 12) = √36 = **6**. ⚠️ But roots do **not** split over addition: **√(a + b) is NOT √a + √b**. For example √(9 + 16) = √25 = 5, whereas √9 + √16 = 3 + 4 = 7. Five is not seven — so never break a root across a plus sign.

Multiply the surds

An interactive activity.

Mind the trap

What is √(9 + 16)?

• 5
• 7
• 25
• 12

Adding like surds ➕

Surds add and subtract like **algebra**: you can only combine **like** surds (the same root). • 3√2 + 5√2 = **8√2** (same √2, add the coefficients) • 7√5 − 2√5 = **5√5** But √2 + √3 will **not** combine — different roots, just as *x + y* stays apart. If surds look unlike, try **simplifying** first: √8 + √2 = 2√2 + √2 = **3√2**.

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.

Rationalising the denominator 🧹

A surd on the bottom of a fraction is untidy, so we **rationalise** — clear it by multiplying top and bottom by that surd (which does not change the value): **1/√2** = (1 × √2) / (√2 × √2) = **√2 / 2**. It often simplifies neatly: **6/√3** = 6√3 / (√3 × √3) = 6√3 / 3 = **2√3**.

Rationalise 6/√3

An interactive activity.

Fractional and negative indices 🧮

Indices follow fixed laws. Two that trip people up: • A **negative** index means **reciprocal**: a⁻ⁿ = 1/aⁿ. • A **fractional** index means a **root**: the bottom is the root, the top is the power — 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**.

Order the working

An interactive activity.

Match each power to its value