CDS Mathematics · Surds, Indices and Simplification

Equations with Surds

A ratio of the form (√A + √B)/(√A − √B) is untangled by componendo and dividendo; other surd equations clear by factorising or isolating one root and squaring.

All Surds, Indices and Simplification notes CDS Mathematics strategy

Why this matters

Seven PYQs, four of them HARD. Almost all have the same shape — a sum of two roots over their difference, equal to something — and componendo and dividendo turns that into √A/√B in one line.

Concept 1 of 2: Componendo and dividendo

\dfrac pq = \dfrac rs

, then

\dfrac{p + q}{p - q} = \dfrac{r + s}{r - s}

. Applied to

\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k

, the sum and difference of top and bottom are

2\sqrt A

and

2\sqrt B

, so the roots separate.

Definition

$\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k$ gives $\dfrac{\sqrt A}{\sqrt B} = \dfrac{k + 1}{k - 1}$ , so $\dfrac AB = \left(\dfrac{k + 1}{k - 1}\right)^2$ .
Alternatively multiply top and bottom by the numerator: the denominator becomes $A - B$ .
After squaring, reject roots that make the original expression undefined, and the trivial $x = 0$ when the question asks for a non-zero value.

Componendo and dividendo

\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k \;\Rightarrow\; \dfrac{\sqrt A}{\sqrt B} = \dfrac{k + 1}{k - 1}

Worked example

Solve

\dfrac{\sqrt{x + 5} + \sqrt{x - 3}}{\sqrt{x + 5} - \sqrt{x - 3}} = 3

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q4Moderate

Example 1 · Surds, Indices and Simplification · Equations with Surds

\frac{\sqrt{x + 20} + \sqrt{x - 1}}{\sqrt{x + 20} - \sqrt{x - 1}} = \frac{7}{3}

, then what is the value of

\sqrt{(x + 20)(x - 1)}

k + 1 over k − 1, not the other way

From

\dfrac{S + D}{S - D} = k

the ratio

\dfrac SD

\dfrac{k + 1}{k - 1}

. Swapping it gives the reciprocal, which is usually an option.

Drill 4 more on componendo and dividendo

Concept 2 of 2: Factorising with square roots

Treat

\sqrt x

and

\sqrt y

as the letters:

x - y

is a difference of squares and

x\sqrt y + y\sqrt x

has the common factor

\sqrt{xy}

. Cancelling leaves a linear relation between

\sqrt x

and

\sqrt y

Definition

$x - y = (\sqrt x - \sqrt y)(\sqrt x + \sqrt y)$ for $x, y \ge 0$ .
$x\sqrt y + y\sqrt x = \sqrt{xy}(\sqrt x + \sqrt y)$ .
$(x + \sqrt{1 + x^2})(-x + \sqrt{1 + x^2}) = 1$ , and $t + \sqrt{1 + t^2}$ always increases with $t$ .

Difference of squares in roots

x - y = (\sqrt x - \sqrt y)(\sqrt x + \sqrt y)

Worked example

Simplify

\dfrac{x - y}{\sqrt x + \sqrt y}

and find its value at

x = 49

y = 16

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q17Hard

Example 2 · Surds, Indices and Simplification · Equations with Surds

\frac{x - y}{x\sqrt{y} + y\sqrt{x}} = \frac{1}{\sqrt{x}}

;

(x > 0,\ y > 0)

then what is the value of

\frac{x}{y}

Square the ratio at the end

\sqrt x = 2\sqrt y

gives

\dfrac xy = 4

, not

2

. The ratio of the roots is the square root of the ratio asked for.

Drill 1 more on factorising with square roots

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)

Componendo and dividendo
Componendo and dividendo
$\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k \;\Rightarrow\; \dfrac{\sqrt A}{\sqrt B} = \dfrac{k + 1}{k - 1}$
Factorising with square roots
Difference of squares in roots
$x - y = (\sqrt x - \sqrt y)(\sqrt x + \sqrt y)$

Watch out for (2)

k + 1 over k − 1, not the other way
→ Componendo and dividendo
Square the ratio at the end
→ Factorising with square roots

Test yourself on Surds, Indices and Simplification

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Surds, Indices and Simplification chapter test Drill all 7 questions