Squares, Cubes and Roots — AFCAT Numerical Ability Formulas and Traps

Squares, Their Last Digits, and Perfect Squares

Squaring a number, and the square identities

Square of a sum or a difference

(a \pm b)^2 = a^2 \pm 2ab + b^2

Testing a perfect square by prime factors

Perfect square test

n = p^{a} q^{b} \cdots \text{ is a square when } a, b, \ldots \text{ are all even}

The last digit of a square

Last digit of n	Last digit of n²	Examples
0	0	$20^2 = 400$
1 or 9	1	$11^2 = 121$ , $19^2 = 361$
2 or 8	4	$12^2 = 144$ , $18^2 = 324$
3 or 7	9	$23^2 = 529$ , $17^2 = 289$
4 or 6	6	$14^2 = 196$ , $16^2 = 256$
5	5	$25^2 = 625$ The last two digits are always 25.

Common traps

The middle term

(4 + 5)^2 = 81

, not

16 + 25 = 41

. The square of a sum keeps the cross term

2ab

.

Brackets decide the sign

(-7)^2 = 49

, but

-7^2 = -49

: without brackets only the 7 is squared.

A good ending is not proof

2,231 ends in 1, a possible ending, but it lies between

47^2 = 2209

and

48^2 = 2304

. The last digit can rule a number out, never in.

Squares ending in 5

2,675 ends in 5, but in 75, not 25, so it is not a perfect square.

One odd power is enough to fail

300 = 2^2 \times 3 \times 5^2

has two pairs, but the single 3 means it is not a perfect square.

Factor all the way down

72 = 8 \times 9

looks like two squares' worth, but

8 = 2^3

. So

72 = 2^3 \times 3^2

has an odd power and is not a square.

Square Roots of Whole Numbers, Decimals and Fractions

Learn this subtopic in the notes

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

Common traps

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.

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.

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.

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

.

Cubes, Cube Roots, and Square and Cube Roots Together

Learn this subtopic in the notes

Cubing a number, and the cube identities

Sum and difference of cubes

a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)

Cube roots by prime factors

Cube root by prime factors

\sqrt[3]{p^{3a} q^{3b}} = p^{a} q^{b}

Cube roots of negative numbers, and cube root equations

Clearing a cube root

\sqrt[3]{ax + b} = k \;\Rightarrow\; ax + b = k^3

Cube roots of decimals and fractions

Cube root of a decimal

\sqrt[3]{\dfrac{a}{10^{3k}}} = \dfrac{\sqrt[3]{a}}{10^{k}}

Square roots and cube roots in one expression

One root at a time

\sqrt{a^2} = a, \quad \sqrt[3]{b^3} = b

The cubes from 1 to 10, and spotting a perfect cube

n	n³	Last digit of n³
1	1	1
2	8	8 2 and 8 swap: a cube ending in 8 has a root ending in 2.
3	27	7 3 and 7 swap: a cube ending in 7 has a root ending in 3.
4	64	4
5	125	5
6	216	6
7	343	3
8	512	2
9	729	9
10	1,000	0

Common traps

An odd power keeps the sign

(-5)^3 = -125

, but

(-5)^2 = 25

. Cubes of negative numbers are negative.

A cube of a sum is not a sum of cubes

(1 + 4)^3 = 125

, not

1 + 64 = 65

. The expansion has two middle terms.

Cubes can end in any digit

A cube can end in 2, 3, 7 or 8:

8^3 = 512

and

7^3 = 343

. The last digit never rules a number out as a cube, unlike a square.

Bracket with cubes, not squares

For 9,261, strike off three digits, not two, and compare 9 with the cubes 8 and 27. Comparing 92 with squares sends you to the wrong tens digit.

Groups of three, not pairs

\sqrt[3]{2^6}

is

2^2 = 4

, not

2^3 = 8

. Divide each exponent by 3 for a cube root, by 2 only for a square root.

Cube, do not square

\sqrt[3]{x} = 4

gives

x = 64

, not 16. Clearing a cube root needs the cube of both sides.

A negative cube root exists

\sqrt[3]{-216} = -6

. Answering 'undefined' mixes up cube roots with square roots.

Decimal places in threes

\sqrt[3]{0.0027}

is not 0.03, because

0.03^3 = 0.000027

. Four decimal places do not give a neat cube root:

\sqrt[3]{0.0027}

is about 0.139.

A third of the places

\sqrt[3]{0.000064} = 0.04

: six places become two. Halving them, as for a square root, gives 0.004.

Read the small index

\sqrt[3]{64} = 4

but

\sqrt{64} = 8

. Missing the 3 on the root sign doubles the answer here.

The Least Number That Makes a Perfect Square or Cube

Learn this subtopic in the notes

Least multiplier or divisor, by prime factors

What a perfect power needs

\text{square: every exponent even}, \quad \text{cube: every exponent a multiple of } 3

Least number to add or subtract for a perfect square

Nearest squares

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

Common traps

Least means only the missing primes

72 = 2^3 \times 3^2

. Multiplying by 8 also gives a square (576), but the least multiplier is 2, giving 144.

A cube needs threes

40 = 2^3 \times 5

. A cube needs two more 5s, so multiply by 25 to reach 1,000. Multiplying by 5 gives

2^3 \times 5^2

, still not a cube.

Read the verb

Dividing removes the leftover primes; multiplying adds the missing ones. For a cube the two answers usually differ, so answer the one asked.

Which neighbour

Adding uses the square above; subtracting uses the square below. For 850 the answers are 50 (up to 900) and 9 (down to 841); swapping the neighbours mixes them up.

Step from a round square

Guessing the root by eye often lands one off. Start from a square you know, such as

30^2 = 900

, and step up or down by

2k + 1

.

Surds: Multiplying, Simplifying and Rationalising

Learn this subtopic in the notes

Multiplying and dividing roots

Product and quotient rules

\sqrt[n]{a}\,\sqrt[n]{b} = \sqrt[n]{ab}, \quad \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac ab}

Simplest form of a surd, and like surds

Taking out a square or cube factor

\sqrt{a^2 b} = a\sqrt b, \quad \sqrt[3]{a^3 b} = a\sqrt[3]{b}

Squaring a two-term surd, and its square root

Square of a two-term surd

(\sqrt a \pm \sqrt b)^2 = a + b \pm 2\sqrt{ab}

Rationalising a denominator

Rationalising with the conjugate

\dfrac{1}{\sqrt a + \sqrt b} = \dfrac{\sqrt a - \sqrt b}{a - b}

Common traps

Products combine, sums do not

\sqrt 3 + \sqrt{12}

is not

\sqrt{15}

. Only a product or a quotient goes under one root sign.

One kind of root at a time

\sqrt 5 \times \sqrt[3]{5}

is neither

\sqrt{25}

nor

\sqrt[3]{25}

. A square root and a cube root combine only as fractional powers, which come on the last page.

Take out the largest square

\sqrt{200} = 2\sqrt{50}

is true but not finished. The largest square factor is 100, so

\sqrt{200} = 10\sqrt 2

.

Unlike surds stay apart

2\sqrt 5 + 3\sqrt 2

cannot be written as

5\sqrt 7

or

5\sqrt{10}

. It is already as simple as it gets.

Keep the cross term

(\sqrt{11} + \sqrt 5)^2

is

16 + 2\sqrt{55}

, not 16. The cross term

2\sqrt{ab}

is the part most often dropped.

Larger first with a minus

A square root of

13 - 2\sqrt{40}

is

\sqrt 8 - \sqrt 5

, that is

2\sqrt 2 - \sqrt 5

. Writing

\sqrt 5 - \sqrt 8

gives a negative number.

Multiply the top too

Multiplying only the bottom changes the value.

\dfrac{1}{\sqrt 6 - 2}

becomes

\dfrac{\sqrt 6 + 2}{2}

only when top and bottom are both multiplied by

\sqrt 6 + 2

.

A cube root needs two more

To clear

\sqrt[3]{3}

, multiply by

\sqrt[3]{9}

, not by

\sqrt[3]{3}

:

\sqrt[3]{3} \times \sqrt[3]{3} = \sqrt[3]{9}

is still a root.

Roots as Fractional Powers, and the Laws of Indices

Learn this subtopic in the notes

A root as a fractional power

Fractional power

a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m

The laws of indices

Laws of indices

a^m a^n = a^{m + n}, \quad \dfrac{a^m}{a^n} = a^{m - n}, \quad (a^m)^n = a^{mn}

Equations with equal powers

Equal powers

a^x = b^y = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y}

Common traps

The bottom is the root

x^{2/3}

is the cube root of

x^2

, not the square root of

x^3

, which is

x^{3/2}

.

A half power is not half

36^{1/2} = 6

, not 18. The power one half takes a square root.

Add, do not multiply

x^{1/2} \times x^{1/4} = x^{3/4}

, not

x^{1/8}

. Multiplying powers of one base adds the indices.

A negative index is not a negative number

5^{-2} = \tfrac{1}{25}

, a positive number, not

-25

.

Same base first

4^x = 2^6

does not give

x = 6

. Write 4 as

2^2

:

2^{2x} = 2^6

, so

x = 3

.

A product becomes a sum

If

a = k^{1/x}

and

b = k^{1/y}

, then

ab = k^{1/x + 1/y}

, not

k^{1/(xy)}

.

Squares, Cubes and Roots formulas

Squares, Their Last Digits, and Perfect Squares

Squaring a number, and the square identities

Testing a perfect square by prime factors

The last digit of a square

Common traps

Square Roots of Whole Numbers, Decimals and Fractions

Square root of a large perfect square

Square roots in an expression

Square roots of decimals and fractions

Equations with a square root

Common traps

Cubes, Cube Roots, and Square and Cube Roots Together

Cubing a number, and the cube identities

Cube roots by prime factors

Cube roots of negative numbers, and cube root equations

Cube roots of decimals and fractions

Square roots and cube roots in one expression

The cubes from 1 to 10, and spotting a perfect cube

Common traps

The Least Number That Makes a Perfect Square or Cube

Least multiplier or divisor, by prime factors

Least number to add or subtract for a perfect square

Common traps

Surds: Multiplying, Simplifying and Rationalising

Multiplying and dividing roots

Simplest form of a surd, and like surds

Squaring a two-term surd, and its square root

Rationalising a denominator

Common traps

Roots as Fractional Powers, and the Laws of Indices

A root as a fractional power

The laws of indices

Equations with equal powers

Common traps

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