GCSE · Maths · AQA · Spec 8300 · Higher

Surds — calculate, simplify, rationalise (Higher)

√12 and 2√3 look like two different numbers. They're the same number in different clothes — and today you'll learn to change the clothes without changing the number.

Same number, different clothes

Where do these five numbers live?

Drag each number to where you think it sits — a good estimate is all you need. Handy anchors: √9 = 3 and √16 = 4, √2 is about 1.4 and √3 is about 1.7. Then check your line and look for any numbers that land on exactly the same spot.

Maths · Algebra

Simplify: find the square hiding inside

Step through each line. Watch what happens to the square factor.

GoalSimplify √12
1
√12
start

Ask: which square numbers (4, 9, 16, 25, 36, …) go into 12? Only 4 does.

2
3
4

Step 1 of 4

Ask: which square numbers (4, 9, 16, 25, 36, …) go into 12? Only 4 does.

Calculate exactly

Multiply every part by every part

Expand (√2 + √3)(√2 − √3). Fill each cell with its row times its column, then collect the cells. Two rules you need: √a × √a = a, and √a × √b = √(ab). Type √ as sqrt if your keyboard has no √ — for example sqrt6.

Grid: each cell is its row times its column
√2−√3
√2
+√3

Type x² as x^2 if you cannot type ². Spaces do not matter.

Rationalise the denominator

Choose what to multiply by

Write each with a whole-number denominator: (a) 10/√5 (b) 4/(3 + √5)

  1. (a) 10/√5A single surd on the bottom. We want a whole number there instead.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Change how a number looks — never what it is.

What you need to know

  • A surd such as √3 is an exact value; a decimal like 1.732 is only an approximation of it.
  • √(ab) = √a × √b, so √a × √a = a. But √(a + b) is NOT √a + √b.
  • Simplify a surd by pulling out its largest square factor: √12 = √(4 × 3) = 2√3.
  • Rationalise k/√b by multiplying top and bottom by √b; rationalise a denominator a + √b or a − √b by multiplying top and bottom by the conjugate (same bracket, opposite sign).
  • Simplifying and rationalising change the form of a number, never its value.

The big picture

A surd is a root, like √2 or √3, that isn't a whole number or a fraction — its decimal never ends or repeats, so the only exact way to write it is with the root sign. You can calculate exactly with surds using √a × √b = √(ab) and √a × √a = a. You simplify a surd by pulling out its largest square factor, as in √12 = √(4 × 3) = √4 × √3 = 2√3. And you rationalise a denominator by multiplying top and bottom by something equal to 1. None of these moves changes the value — only how it's written.

Key points

1√a × √b = √(ab), and √a × √a = a.
2A root splits over × but never over +.
3Largest square factor first — then you finish in one pass.
4(a + √b)(a − √b) = a² − b: the surd terms cancel.
5Like surds collect like algebra: 2√2 + 3√2 = 5√2.

Worked example

Problem

Show that (√8 + √18)² = 50.

⚠ Watch out

Splitting a root over a sum. √(36 + 64) = √100 = 10, not √36 + √64 = 14. The root splits over multiplication only.

🧠

Memory hook

Find the square hiding inside; multiply by 1 in disguise.

✓

Check yourself

Without a calculator: simplify √45 fully, then write 6/√3 with a whole-number denominator.

Flashcards

(6)
What makes a surd exact where a decimal isn't?
A surd like √2 has a decimal that never ends or repeats, so writing it with the root sign is the only exact form. Any decimal is an approximation.
√a × √b = ?
√(ab). It works for multiplying — never for adding: √(a + b) is not √a + √b.
√a × √a = ?
a — a whole number when a is a whole number. This is what makes rationalising possible.
How do you know a surd is fully simplified?
The number under the root has no square factor left (other than 1). Pull out the LARGEST square factor to get there in one pass.
To rationalise k/√b, multiply by…?
√b/√b — top and bottom — which is multiplying by 1.
To rationalise a denominator a + √b, multiply by…?
The conjugate: (a − √b)/(a − √b). Then (a + √b)(a − √b) = a² − b, with no surd left.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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