CDS Mathematics · Surds, Indices and Simplification

Simplifying Surd Expressions

When x is built from cube roots, name the cube root and use its cube; when an expression holds square roots of squares, remember √(a²) = |a|.

All Surds, Indices and Simplification notes CDS Mathematics strategy

Why this matters

Six PYQs, three of them HARD. The hard ones all use one idea: if u = ∛2, then u³ = 2, so any polynomial in x = 2 + u + u² reduces to a number. The rest are careful expansions.

Concept 1 of 2: Expressions built from cube roots

x

made of

\sqrt[3]{2}

and its square looks hopeless to cube directly. Move the plain number to the other side, cube both sides, and replace every

u^3

2

: the surds either cancel or rebuild

x

itself.

Definition

Let $u = \sqrt[3]{a}$ , so $u^3 = a$ .
If $x - c = u + u^2$ , then $(x - c)^3 = u^3(1 + u)^3 = a(1 + 3u + 3u^2 + u^3)$ , which is $a(1 + a) + 3a(x - c)$ .
Expand the cubic asked for and compare.
For square roots, $x = 2 + 3\sqrt2$ satisfies $(x - 2)^2 = 18$ , a quadratic.

Cube with u³ = a

(u + u^2)^3 = u^3(1 + u)^3 = a\,(1 + u)^3

Worked example

x = 1 + \sqrt[3]{3} + \sqrt[3]{9}

, find

x^3 - 3x^2 - 6x

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q3Hard

Example 1 · Surds, Indices and Simplification · Simplification of Expressions

x = 2 + 2^{2/3} + 2^{1/3}

, then the value of the expression

x^3 - 6x^2 + 6x

will be

Move the whole number before cubing

Cubing

x = 2 + u + u^2

directly produces dozens of terms. Cubing

x - 2 = u + u^2

produces four, because

u^3

factors out.

Drill 2 more on expressions built from cube roots

Concept 2 of 2: Square roots of squares and products of sums

\sqrt{a^2}

is the size of

a

, never negative. And a product of four brackets like

(a + b + c)(a + b - c)\ldots

pairs up into differences of squares.

Definition

$\sqrt{a^2} = |a|$ , so $\sqrt{(a - b)^2} + \sqrt{(b - a)^2} = 2|a - b|$ .
$(a + b + c)(a + b - c) = (a + b)^2 - c^2$ .
$(a + b + c)(-a + b + c)(a - b + c)(a + b - c) = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 - a^4 - b^4 - c^4$ .
Mixed numbers under a root: convert first, $3\tfrac{3}{8} = \dfrac{27}{8}$ .

Root of a square

\sqrt{a^2} = |a|

Worked example

Evaluate

(\sqrt2 + \sqrt3 + 1)(\sqrt2 + \sqrt3 - 1)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q19Hard

Example 2 · Surds, Indices and Simplification · Simplification of Expressions

What is

(\sqrt{2} + \sqrt{3} + \sqrt{5})(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} + \sqrt{5})(-\sqrt{2} + \sqrt{3} + \sqrt{5})

equal to ?

√(a²) is not a

\sqrt{(a - b)^2} = |a - b|

, which is

b - a

when

b > a

. Writing

a - b

makes a positive quantity look negative.

Drill 2 more on square roots of squares and products of sums

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Expressions built from cube roots
Cube with u³ = a
$(u + u^2)^3 = u^3(1 + u)^3 = a\,(1 + u)^3$
Square roots of squares and products of sums
Root of a square
$\sqrt{a^2} = |a|$

Watch out for (2)

Move the whole number before cubing
→ Expressions built from cube roots
√(a²) is not a
→ Square roots of squares and products of sums

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