Pull out the perfect squares, then clear the root off the bottom.
Simplify
√72
72 = 36 × 2, and 36 is the largest perfect square inside, so √72 = 6√2. A is only half-simplified — √18 still hides a factor of 9. Always pull out the biggest square available, or you will be simplifying twice.
72 = 36 × 2
36 is a perfect square
√72 = √36 · √2
roots split across ×
= 6√2
≈ 8.49
Starting with 4 × 18 gives 2√18, then 2 · 3√2 = 6√2. Same answer, one extra step. Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100.
√9 · 4 = 3 · 2 = 6 ✓ but √9 + 16 = √25 = 5, not 3 + 4 = 7.
Treat √2 like a letter — 3√2 + 5√2 is just 8 of them.
3√2 + 5√2 = 8√2
√2 + √3
leave it alone
√8 + √18
looks unlike
= 2√2 + 3√2
simplify first
= 5√2
now they match
Always simplify before deciding two radicals will not combine.
Multiply top and bottom by that root — which is multiplying by 1.
6√3 · √3√3
√3 ÷ √3 = 1
= 6√33
√3 · √3 = 3
= 2√3
6 ÷ 3 = 2
The value never changed — 6√3 and 2√3 are both about 3.46. Only the shape changed, and exam answers are written in the second shape.
Flip the middle sign, and the difference of squares kills the root.
4√5 − 1 · √5 + 1√5 + 1
the conjugate
bottom = (√5)² − 1² = 5 − 1
the middle terms cancel
= 4(√5 + 1)4
bottom is 4
= √5 + 1
the 4s cancel
The conjugate of √5 − 1 is √5 + 1, not −√5 − 1 or 1 − √5. Change the middle sign and nothing else.
A number in front gets squared too: (3√5 − 2)(3√5 + 2) = (3√5)² − 2² = 9 × 5 − 4 = 41. And a sum over a conjugate: 1√6 + √5 = √6 − √56 − 5 = √6 − √5.
| Tempting | Correct | Why |
|---|---|---|
| √a + b = √a + √b | false | √25 ≠ 3 + 4 |
| √ab = √a · √b | true | products are fine |
| √x² = x | |x| | √−3² = 3, not −3 |
| (√a)² = a | true | for a ≥ 0 |
| √2 · √8 = √16 | = 4 | a surd can vanish |
A radical answer is only finished when no perfect square is left inside and no root is left underneath.
Simplify √512.
√512 = √5√12 = √52√3
split, then simplify the bottom
√5 · √32√3 · √3 = √156
multiply by √3 only, not √12
Different: simplify the bottom root first, so the number you rationalise with is as small as possible (≈ 0.645 both ways).
Simplify 3√−54.
−54 = (−27) × 2
largest perfect cube
3√−27 · 3√2 = −33√2
(−3)³ = −27
Different: cube roots look for 8, 27, 64, 125 — and a negative inside is fine, because an odd power keeps its sign. √−54 has no real value; 3√−54 does.
Expand (√5 + √3)².
(√5)² + 2(√5)(√3) + (√3)²
(a + b)² = a² + 2ab + b²
5 + 2√15 + 3 = 8 + 2√15
≈ 15.75, and (2.236 + 1.732)² ≈ 15.75 ✓
Different: squaring a sum of surds leaves a surd in the middle term; only the conjugate product clears it.
Squaring each term separately drops the 2√15. A square of a sum always has a middle term — with surds, exactly as with letters.
Letters under the root: pull out even powers, exactly as with numbers. Simplify √(27a5b²) for a, b > 0.
27a5b² = (9a4b²)(3a)
split off the largest perfect square
√(9a4b²) = 3a²b
halve each even exponent
√(27a5b²) = 3a²b√3a
odd leftovers stay inside
A surd as the value of x: find x² + 4x when x = √5 − 2. Rearranging first saves the expanding.
x² + 4x = (x + 2)² − 4
complete the square
x + 2 = √5, so (x + 2)² = 5
the surd squares away
x² + 4x = 5 − 4 = 1
direct: (9 − 4√5) + (4√5 − 8) = 1 ✓
A root inside a root: simplify √11 − 6√2. Look for (p − √q)² = p² + q − 2p√q.
2p√q = 6√2 → p√q = 3√2
try p = 3, q = 2
p² + q = 9 + 2 = 11 ✓
the outside number matches
√11 − 6√2 = 3 − √2
larger term first, so the root is positive
A root does not split across a minus (Step 5). The only way in is to recognise the inside as a perfect square, and √2 − 3 is wrong too: it is negative, and a square root never is.
Radical items are simplify, rationalise or “evaluate at x = a surd”. The choices are all written in finished form, so finish yours the same way. The quickest check is the decimal one below: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 — compare your expression with the choices to two decimal places.
Simplify
Simplify √50 + √32
√50 = 5√2 and √32 = 4√2. Once they share the same surd you can add the counts: 9√2. A adds under the root, but √a + √b is never √a + b. Simplify first, then check whether the surds match.
