AFCAT Numerical Ability · Decimal and Fractions

Squares, Cubes and Their Roots, Including Decimals and Nested Roots

Know the squares and cubes by heart, find the nearest perfect square, take roots of decimals by halving or thirding the places, and open nested roots from the inside.

All Decimal and Fractions notes

Why this matters

AFCAT asks from this page more than from any other in the chapter: seven of its past-year questions are here, from 2016 to 2019. Most are quick if the squares and cubes are known by heart. The marks go on the decimal places in a root, and on opening a nested root from the outside instead of the inside.

Concept 1 of 3: Perfect squares and cubes, and the nearest square

Every number sits between two perfect squares. The least number to add reaches the square above it; the least number to subtract drops to the square below.

Definition

Know the squares up to $30^2 = 900$ and the cubes up to $15^3 = 3375$ ; they make most roots one step.
If $n^2 \le N < (n + 1)^2$ : add $(n + 1)^2 - N$ to reach a square, or subtract $N - n^2$ .
Rows of a square: if $N$ people form a square with some left over, the square holds the largest perfect square below $N$ .
The largest $k$ -digit square is the square of the largest root that keeps $k$ digits.
Take square factors out of a root: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt2$ .

Reaching the nearest square

n^2 \le N < (n + 1)^2: \quad \text{add } (n + 1)^2 - N \text{ or subtract } N - n^2

Worked example

Find the least number to add to 1,000 to make a perfect square, and the least number to subtract.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 01-314Easy

Example 1 · Decimal and Fractions · Squares, Cubes and Roots

What is the least number to be added to 7700 to make it a perfect square?

Add uses the square above, subtract the square below

For 300, the squares around it are

17^2 = 289

and

18^2 = 324

. The least number to add is 24; the least to subtract is 11. Mixing them up gives the other one.

Find the range from the number of digits

The greatest 5-digit square is

316^2 = 99856

, not

99^2

, which has only four digits.

Drill 7 more on perfect squares and cubes, and the nearest square

Concept 2 of 3: Square and cube roots of decimals

Squaring a number doubles its decimal places:

0.03^2 = 0.0009

. So a square root halves them, and a cube root divides them by 3.

Definition

A decimal has an exact square root only if its number of decimal places is even; the root has half as many.
For a cube root the places must be a multiple of 3; the root has a third as many.
Moving the point two places in the number moves it one place in the square root: if $\sqrt{2116} = 46$ , then $\sqrt{21.16} = 4.6$ .
When the root is not exact, estimate from the nearest squares and pick the closest option.

Roots of decimals

\sqrt{\dfrac{a}{10^{2n}}} = \dfrac{\sqrt a}{10^n}, \qquad \sqrt[3]{\dfrac{a}{10^{3n}}} = \dfrac{\sqrt[3]{a}}{10^n}

Worked example

Find

\sqrt{0.0049} + \sqrt[3]{0.027}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 01-303Moderate

Example 2 · Decimal and Fractions · Squares, Cubes and Roots

\sqrt{18225} = 135

, then the value of

\sqrt{18225} + \sqrt{182.25} + \sqrt{1.8225} + \sqrt{0.018225}

An odd number of places has no neat root

\sqrt{0.09} = 0.3

, but

\sqrt{0.9}

is about

0.949

. The

0.3

option for

\sqrt{0.9}

is a trap.

Halve the places, do not keep them

\sqrt{0.0016} = 0.04

, with two places. Writing

0.004

keeps three and is ten times too small.

Drill 4 more on square and cube roots of decimals

Concept 3 of 3: Nested and never-ending roots

A root inside a root inside a root is opened like a set of boxes, from the innermost outwards. A root that goes on for ever contains a copy of itself, so call it

x

and write one equation.

Definition

Finite nesting: evaluate the innermost root, add it to the number beside it, and take the next root. Each step is usually a perfect square.
$x = \sqrt{a + \sqrt{a + \sqrt{a + \dots}}}$ : the part under the first root is $a + x$ , so $x^2 = a + x$ . Keep the positive root.
$x = \sqrt{a\sqrt{a\sqrt{a \dots}}}$ : $x^2 = ax$ , so $x = a$ .

A never-ending root

x = \sqrt{a + \sqrt{a + \dots}} \;\Rightarrow\; x^2 = a + x

Worked example

Find

\sqrt{28 + \sqrt{50 + \sqrt{176 + \sqrt{400}}}}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 01-305Moderate

Example 3 · Decimal and Fractions · Squares, Cubes and Roots

The simplified value of

\sqrt{5 + \sqrt{11 + \sqrt{19 + \sqrt{29 + \sqrt{49}}}}}

Start at the innermost root

\sqrt{90 + \sqrt{100}}

the

\sqrt{100}

comes first. A root does not split over a sum:

\sqrt{36 + 64} = 10

, not

6 + 8

Keep the positive root

x^2 - x - 2 = 0

has roots 2 and

-1

. A square root is never negative, so the answer is 2.

Drill 3 more on nested and never-ending 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 (3)

Perfect squares and cubes, and the nearest square
Reaching the nearest square
$n^2 \le N < (n + 1)^2: \quad \text{add } (n + 1)^2 - N \text{ or subtract } N - n^2$
Square and cube roots of decimals
Roots of decimals
$\sqrt{\dfrac{a}{10^{2n}}} = \dfrac{\sqrt a}{10^n}, \qquad \sqrt[3]{\dfrac{a}{10^{3n}}} = \dfrac{\sqrt[3]{a}}{10^n}$
Nested and never-ending roots
A never-ending root
$x = \sqrt{a + \sqrt{a + \dots}} \;\Rightarrow\; x^2 = a + x$

Watch out for (6)

Add uses the square above, subtract the square below
→ Perfect squares and cubes, and the nearest square
Find the range from the number of digits
→ Perfect squares and cubes, and the nearest square
An odd number of places has no neat root
→ Square and cube roots of decimals
Halve the places, do not keep them
→ Square and cube roots of decimals
Start at the innermost root
→ Nested and never-ending roots
Keep the positive root
→ Nested and never-ending roots

Drill the questions on this topic

17 questions from the bank, with cart and Word export.

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