AFCAT Numerical Ability · Squares, Cubes and Roots

Square Roots of Whole Numbers, Decimals and Fractions

Find the square root of a large perfect square by bracketing it, take roots of decimals and fractions, follow the rules for roots in an expression, and solve equations with a root.

All Squares, Cubes and Roots notes

Why this matters

Half the page is quick roots of perfect squares, and some come dressed as trees planted in rows or paise collected from members. Marks are lost on decimals, where the root has half as many decimal places, and on treating the root of a sum as the sum of the roots.

Concept 1 of 4: Square root of a large perfect square

The root of a four or five digit perfect square has two or three digits. The front of the number fixes the tens, the last digit leaves two choices for the units, and one more square decides between them.

Definition

Strike off the last two digits. The largest square at or below what is left gives the tens: for 5,329, 53 lies between $7^2 = 49$ and $8^2 = 64$ , so the root is in the 70s.
The last digit leaves two choices for the units digit (the table on the previous page, read backwards): a 9 comes from 3 or 7, so 73 or 77.
Compare with the square of the middle: $75^2 = 5625$ is above 5,329, so the root is the smaller choice, 73.
'As many rows as plants in a row' or 'as many paise from each member as there are members' means the total is $n^2$ . The answer is the square root of the total, in the right unit.

Root by bracketing

(10a)^2 < N < (10a + 10)^2 \;\Rightarrow\; 10a < \sqrt N < 10a + 10

Worked example

Find

\sqrt{7056}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Squares, Cubes and Roots · Square Roots

A gardener plants 17956 trees in such a way that there are as many rows as there are trees in a row. The number of trees in a row are

Two candidates for the units digit

2,116 ends in 6, so its root ends in 4 or 6.

45^2 = 2025

is below it, so the root is 46, not 44.

Convert rupees to paise

₹20.25 collected as 'as many paise each as members' is 2,025 paise, so there are 45 members. Taking the root of 20.25 gives 4.5, which cannot be a count.

Drill 4 more on square root of a large perfect square

Concept 2 of 4: Square roots in an expression

A square root undoes a square. It splits over a product or a quotient, but never over a sum. Find each root first, then do the arithmetic.

Definition

$\sqrt{ab} = \sqrt a \times \sqrt b$ and $\sqrt{\tfrac ab} = \dfrac{\sqrt a}{\sqrt b}$ for positive $a$ and $b$ .
$\sqrt{a + b}$ is not $\sqrt a + \sqrt b$ : $\sqrt{36 + 64} = 10$ , but $6 + 8 = 14$ .
$(\sqrt a)^2 = a$ . The root sign means the positive root, so $\sqrt{x^2} = |x|$ : $\sqrt{(-5)^2} = 5$ .
Roots inside roots are worked from the inside out: $\sqrt{\sqrt{625}} = \sqrt{25} = 5$ .

Roots of products and quotients

\sqrt{ab} = \sqrt a\,\sqrt b, \quad \sqrt{\tfrac ab} = \dfrac{\sqrt a}{\sqrt b}

Worked example

Find

\sqrt{40 + \sqrt{70 + \sqrt{121}}}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Squares, Cubes and Roots · Square Roots

Find the value of

\sqrt{343 + \sqrt{307 + \sqrt{273 + \sqrt{241 + \sqrt{225}}}}}

No splitting over a sum

\sqrt{25 + 144} = \sqrt{169} = 13

, not

5 + 12 = 17

. Add first, then take the root.

The root sign is positive

\sqrt{49} = 7

, never

-7

, even though

(-7)^2 = 49

. So

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

, which is

b - a

when

b

is the larger.

Drill 9 more on square roots in an expression

Concept 3 of 4: Square roots of decimals and fractions

Count the decimal places. A decimal with a neat square root has an even number of them, and its root has half as many. A fraction's root is the root of the top over the root of the bottom.

Definition

Write the decimal over 100, 10000, ...: $\sqrt{0.49} = \sqrt{\tfrac{49}{100}} = \tfrac{7}{10} = 0.7$ .
Each pair of decimal places in the number gives one place in the root: $\sqrt{0.0016} = 0.04$ .
Moving the decimal point two places in the number moves it one place in the root: $\sqrt{169} = 13$ , $\sqrt{1.69} = 1.3$ , $\sqrt{0.0169} = 0.13$ .
$\sqrt{\tfrac{25}{36}} = \tfrac{5}{6}$ . A root like this is a rational number.

Root of a decimal

\sqrt{\dfrac{a}{10^{2k}}} = \dfrac{\sqrt a}{10^{k}}

Worked example

Find

\sqrt{1.44} + \sqrt{0.0144}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Squares, Cubes and Roots · Square Roots

What is the value of

\sqrt{29.16} + \sqrt{0.2916} + \sqrt{0.002916} + \sqrt{0.00002916}

An odd number of decimal places

\sqrt{0.9}

is not 0.3, because

0.3^2 = 0.09

. With an odd number of places, write

0.9 = 0.90

: its root is about 0.949.

Halve the places, do not keep them

\sqrt{0.000004} = 0.002

: six places become three. Writing 0.000002 keeps all six.

Drill 5 more on square roots of decimals and fractions

Concept 4 of 4: Equations with a square root

To free

x

from under a root, get the root alone on one side and square both sides. The root undoes, and a plain equation is left.

Definition

$\sqrt x = k$ means $x = k^2$ , and $k$ must be 0 or positive.
$\sqrt{ax + b} = k$ : squaring gives $ax + b = k^2$ . Solve that.
A fraction squares top and bottom: $\sqrt{\tfrac{x}{25}} = \tfrac 35$ gives $\tfrac{x}{25} = \tfrac{9}{25}$ , so $x = 9$ .
If the root has a number in front or beside it, clear that first: $2\sqrt x + 1 = 15$ gives $\sqrt x = 7$ .

Clearing a root

\sqrt{ax + b} = k \;\Rightarrow\; ax + b = k^2

Worked example

Solve

\sqrt{3x - 2} = 5

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Squares, Cubes and Roots · Square Roots

Solve:

\sqrt{x + 3} = 4

Square the whole side

\sqrt{x + 7} = 6

gives

x + 7 = 36

, so

x = 29

. Squaring only the

x

inside gives a wrong equation.

A root is never negative

\sqrt x = -3

has no solution:

x = 9

gives

\sqrt 9 = 3

, not

-3

Drill 4 more on equations with a square root

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 (4)

Square root of a large perfect square
Root by bracketing
$(10a)^2 < N < (10a + 10)^2 \;\Rightarrow\; 10a < \sqrt N < 10a + 10$
Square roots in an expression
Roots of products and quotients
$\sqrt{ab} = \sqrt a\,\sqrt b, \quad \sqrt{\tfrac ab} = \dfrac{\sqrt a}{\sqrt b}$
Square roots of decimals and fractions
Root of a decimal
$\sqrt{\dfrac{a}{10^{2k}}} = \dfrac{\sqrt a}{10^{k}}$
Equations with a square root
Clearing a root
$\sqrt{ax + b} = k \;\Rightarrow\; ax + b = k^2$

Watch out for (8)

Two candidates for the units digit
→ Square root of a large perfect square
Convert rupees to paise
→ Square root of a large perfect square
No splitting over a sum
→ Square roots in an expression
The root sign is positive
→ Square roots in an expression
An odd number of decimal places
→ Square roots of decimals and fractions
Halve the places, do not keep them
→ Square roots of decimals and fractions
Square the whole side
→ Equations with a square root
A root is never negative
→ Equations with a square root

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