AFCAT Numerical Ability · Indices and Surds

Surds: Simplest Form, Adding, Multiplying and Dividing

Know what makes a root a surd, pull the largest square out of the root, combine like surds, and multiply or divide roots under one sign.

All Indices and Surds notes

Why this matters

This is the biggest page of the chapter and the gentlest: only two of its questions are HARD. Nearly all of them come down to one move, taking the largest perfect square out of the root. Marks go when a smaller square is taken out and the answer is left half simplified, or when two unlike surds are added as if they were like.

Concept 1 of 4: What a surd is: order, pure and mixed, like and unlike

A surd is a root that does not come out exact.

\sqrt{49}

is just 7, but

\sqrt{7}

is a never-ending decimal, so it stays as a root. These few words are what the questions use to describe surds.

Definition

A root is a surd only when its value is irrational. Like surds are judged after each surd is in simplest form, which is the next concept on this page.

Term	Meaning	Example
Surd	A root whose value is not a whole number or a fraction	$\sqrt{7}$ , $\sqrt[3]{10}$ ; not $\sqrt{64} = 8$
Order	The index of the root	$\sqrt[3]{6}$ has order 3; $\sqrt{6}$ has order 2
Pure surd	The whole number sits under the root	$\sqrt{12}$
Mixed surd	A rational number times a surd	$2\sqrt{3}$
Like surds	The same number under the same root, once simplified	$4\sqrt{7}$ and $9\sqrt{7}$ ; $\sqrt{12}$ and $\sqrt{75}$ $\sqrt{12} = 2\sqrt 3$ and $\sqrt{75} = 5\sqrt 3$ , so they are like surds even though they look different.
Comparing	Same order: the larger number under the root is larger	$\sqrt{6} < \sqrt{7} < \sqrt{11}$ Different orders: raise both to the LCM of the indices. $\sqrt 2$ and $\sqrt[3]{3}$ to the 6th power give 8 and 9, so $\sqrt[3]{3}$ is larger.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Indices and Surds · Radicals and Surds

Which is a third-order surd?

The root of a perfect square is not a surd

\sqrt{81} = 9

and

\sqrt[3]{64} = 4

are rational. A root is a surd only when it does not come out exact.

The order is the index, not the number inside

\sqrt[3]{100}

is a surd of order 3. The 100 under the root has nothing to do with its order.

Drill 3 more on what a surd is: order, pure and mixed, like and unlike

Concept 2 of 4: Simplest form: take out the largest square

\sqrt{ab} = \sqrt a \times \sqrt b

, so a square factor inside the root can come out as a whole number.

\sqrt{245} = \sqrt{49 \times 5}

, and the 49 comes out as 7.

Definition

Pure to mixed: split the number as (largest perfect square) × (the rest) and take the square root of the square.
Mixed to pure: square the number outside and multiply it in: $k\sqrt a = \sqrt{k^2 a}$ .
A surd is in simplest form when no square factor other than 1 is left under the root.
Decimals: write the decimal as a fraction over a power of 10. $\sqrt{0.0049} = \sqrt{\tfrac{49}{10000}} = \tfrac{7}{100}$ . The answer has half as many decimal places.

Taking a square out

\sqrt{k^2 a} = k\sqrt{a}

Worked example

Write

\sqrt{245}

in simplest form.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Indices and Surds · Radicals and Surds

Simplify:

\sqrt{108}

Take out the LARGEST square

\sqrt{288} = \sqrt{4 \times 72} = 2\sqrt{72}

is not finished, because 72 still holds a square.

288 = 144 \times 2

, so

\sqrt{288} = 12\sqrt 2

A decimal root halves the decimal places

\sqrt{0.0004} = 0.02

, not

0.2

. Four decimal places under the root give two in the answer.

Drill 9 more on simplest form: take out the largest square

Concept 3 of 4: Adding and subtracting like surds

4\sqrt 3 + 2\sqrt 3

is 4 of something plus 2 of the same thing, so it is 6 of them:

6\sqrt 3

. Surds add like

4x + 2x

. Unlike surds, like

x

and

y

, cannot be joined.

Definition

Put every surd in simplest form first.
Add or subtract the numbers in front of the like surds; the root stays as it is.
A number already in front multiplies whatever comes out of the root: $3\sqrt 8 = 3 \times 2\sqrt 2 = 6\sqrt 2$ .
Unlike surds stay as separate terms.

Like surds

m\sqrt a + n\sqrt a = (m + n)\sqrt a

Worked example

Simplify

\sqrt{20} + \sqrt{80} - \sqrt{45}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Indices and Surds · Radicals and Surds

Simplify:

2\sqrt{12} + 3\sqrt{27}

The root does not add

4\sqrt 3 + 2\sqrt 3 = 6\sqrt 3

, not

6\sqrt 6

. Only the numbers in front add.

The square root of a sum is not the sum of the roots

\sqrt 9 + \sqrt{16} = 3 + 4 = 7

, but

\sqrt{9 + 16} = \sqrt{25} = 5

. Never add the numbers under two roots.

Drill 6 more on adding and subtracting like surds

Concept 4 of 4: Multiplying and dividing surds

Roots of the same order multiply and divide under one root sign:

\sqrt a \times \sqrt b = \sqrt{ab}

. The numbers outside the roots multiply among themselves, and the result is then put in simplest form.

Definition

$\sqrt a \times \sqrt b = \sqrt{ab}$ and $\dfrac{\sqrt a}{\sqrt b} = \sqrt{\dfrac ab}$ .
$m\sqrt a \times n\sqrt b = mn\sqrt{ab}$ : outside numbers with outside numbers, roots with roots.
$\sqrt a \times \sqrt a = (\sqrt a)^2 = a$ .
A root of a fraction: simplify the fraction first, or simplify the two roots and cancel the common surd.
The same root on top and bottom cancels, even when the number inside is a letter expression.

One root sign

m\sqrt a \times n\sqrt b = mn\sqrt{ab}, \qquad \dfrac{\sqrt a}{\sqrt b} = \sqrt{\dfrac ab}

Worked example

Simplify

3\sqrt 2 \times 4\sqrt 6

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Indices and Surds · Radicals and Surds

Find:

\sqrt{\frac{8}{18}}

A root times itself is the number inside

\sqrt 7 \times \sqrt 7 = 7

, not

49

and not

\sqrt{14}

Finish in simplest form

2\sqrt 5 \times 3\sqrt 5 = 6 \times 5 = 30

. Stopping at

6\sqrt{25}

leaves an answer that is not among the options.

Drill 8 more on multiplying and dividing surds

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 (3)

Simplest form: take out the largest square
Taking a square out
$\sqrt{k^2 a} = k\sqrt{a}$
Adding and subtracting like surds
Like surds
$m\sqrt a + n\sqrt a = (m + n)\sqrt a$
Multiplying and dividing surds
One root sign
$m\sqrt a \times n\sqrt b = mn\sqrt{ab}, \qquad \dfrac{\sqrt a}{\sqrt b} = \sqrt{\dfrac ab}$

Reference tables (1)

What a surd is: order, pure and mixed, like and unlike6 rows

Term	Meaning	Example
Surd	A root whose value is not a whole number or a fraction	$\sqrt{7}$ , $\sqrt[3]{10}$ ; not $\sqrt{64} = 8$
Order	The index of the root	$\sqrt[3]{6}$ has order 3; $\sqrt{6}$ has order 2
Pure surd	The whole number sits under the root	$\sqrt{12}$
Mixed surd	A rational number times a surd	$2\sqrt{3}$
Like surds	The same number under the same root, once simplified	$4\sqrt{7}$ and $9\sqrt{7}$ ; $\sqrt{12}$ and $\sqrt{75}$ $\sqrt{12} = 2\sqrt 3$ and $\sqrt{75} = 5\sqrt 3$ , so they are like surds even though they look different.
Comparing	Same order: the larger number under the root is larger	$\sqrt{6} < \sqrt{7} < \sqrt{11}$ Different orders: raise both to the LCM of the indices. $\sqrt 2$ and $\sqrt[3]{3}$ to the 6th power give 8 and 9, so $\sqrt[3]{3}$ is larger.

Watch out for (8)

The root of a perfect square is not a surd
→ What a surd is: order, pure and mixed, like and unlike
The order is the index, not the number inside
→ What a surd is: order, pure and mixed, like and unlike
Take out the LARGEST square
→ Simplest form: take out the largest square
A decimal root halves the decimal places
→ Simplest form: take out the largest square
The root does not add
→ Adding and subtracting like surds
The square root of a sum is not the sum of the roots
→ Adding and subtracting like surds
A root times itself is the number inside
→ Multiplying and dividing surds
Finish in simplest form
→ Multiplying and dividing surds

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