Surds, Indices and Simplification — SSC CGL Formulas and Traps

BODMAS and Brackets

The order of operations (BODMAS)

Order of operations

\text{Brackets} \to \text{Orders} \to \div,\ \times\ (\text{left to right}) \to +,\ -\ (\text{left to right})

Removing nested brackets from the inside out

Minus before a bracket

a - (b - c) = a - b + c \qquad a - (-b) = a + b

The word 'of' and the bar in BODMAS

Of against times after a division

a \div b \text{ of } c = \frac{a}{bc} \qquad a \div b \times c = \frac{ac}{b}

Common traps

Multiplying before dividing

In

36 \div 6 \times 2

the division is on the left, so it goes first:

6 \times 2 = 12

. Doing

6 \times 2

first gives

36 \div 12 = 3

, which is wrong. Division and multiplication share one level and go left to right.

Adding before subtracting

20 - 5 + 3

is

15 + 3 = 18

, not

20 - 8 = 12

. Addition and subtraction also share one level and go left to right.

Flipping only the first sign after a minus

30 - (12 - 5 + 2)

is

30 - 12 + 5 - 2 = 21

. A minus before a bracket flips every sign inside it. Flipping only the 12 gives

30 - 12 - 5 + 2 = 15

, a typical wrong option.

Losing a negative bracket

In

4 - (6 - 6 - 8)

the bracket is

-8

, so the value is

4 + 8 = 12

. Writing

4 - 8 = -4

drops the minus that the bracket carries.

Treating 'of' as an ordinary multiplication

48 \div 8 \text{ of } 2

is

48 \div 16 = 3

. Reading 'of' as a plain

\times

gives

48 \div 8 \times 2 = 12

, and SSC prints that as a wrong option.

Ignoring the bar

A bar over

4 - 9

makes it one number,

-5

, so

10 - \overline{4 - 9} = 10 + 5 = 15

. Without the bar the minus would touch only the 4, giving

10 - 4 - 9 = -3

.

Doing 'of' before its own bracket

In

3 \text{ of } (2 + 5)

the bracket still comes first:

3 \times 7 = 21

. 'Of' outranks division, not brackets.

Fractions and Decimals

Learn this subtopic in the notes

Fractions and mixed numbers in BODMAS

Fraction rules

\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \qquad \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad p\frac{q}{r} = \frac{pr + q}{r}

Working with decimals

Recurring decimals and handy divisions

0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99} \qquad x \div 0.25 = 4x \qquad x \div 0.5 = 2x

$a$ the single repeating digit
$ab$ the two-digit number made by the repeating pair

Comparing and ordering fractions

Cross-multiplication test

\frac{a}{b} > \frac{c}{d} \text{ exactly when } ad > bc \quad (b, d > 0)

Continued fractions

Building one level

a + \frac{1}{b} = \frac{ab + 1}{b} \qquad \frac{1}{\;p/q\;} = \frac{q}{p}

Common traps

Multiplying mixed numbers part by part

2\frac{1}{2} \times 3\frac{1}{3}

is not

6\frac{1}{6}

. Convert first:

\frac{5}{2} \times \frac{10}{3} = \frac{25}{3} = 8\frac{1}{3}

.

Turning the wrong fraction upside down

Only the divisor is flipped:

\frac{2}{5} \div \frac{4}{15} = \frac{2}{5} \times \frac{15}{4} = \frac{3}{2}

. Flipping the first fraction instead gives

\frac{5}{2} \times \frac{4}{15} = \frac{2}{3}

, the reciprocal of the answer.

Too few decimal places in a product

0.3 \times 0.2 = 0.06

, not 0.6. Each number has one decimal place, so the product needs two.

Expecting division to make a number smaller

Dividing by a number below 1 makes the result bigger:

4.5 \div 0.5 = 9

. An option smaller than 4.5 can be ruled out at once.

Judging a fraction by the size of its numbers

\frac{9}{70}

has bigger numbers than

\frac{2}{3}

but is far smaller: about 0.13 against 0.67. Compare values, not the size of the top and bottom.

Starting a continued fraction from the top

In

2 + \frac{1}{3 + \frac{1}{2}}

you cannot add 2 and 1 first. The bottom gives

3 + \frac{1}{2} = \frac{7}{2}

, its reciprocal is

\frac{2}{7}

, and the value is

2 + \frac{2}{7} = \frac{16}{7}

.

Missing Numbers and Letters

Learn this subtopic in the notes

Simplifying and evaluating expressions in letters

Powers of a negative number

(-a)^{2} = a^{2} \qquad (-a)^{3} = -a^{3}

Common traps

Pulling the unknown away from its multiplier

In

20 - 3 \times y = 5

the 3 belongs to

y

:

3y = 15

, so

y = 5

. Working

20 - 3 = 17

first breaks BODMAS and gives

y = \frac{5}{17}

.

Skipping the check

An answer found by undoing steps is easily off by a sign. Put it back into the original line before marking it: the check takes a few seconds and catches the slip that the wrong options are built on.

Squaring a negative without brackets

-3^2

means

-(3^2) = -9

, but

(-3)^2 = 9

. When you put

x = -3

into

x^2

, write

(-3)^2

.

Adding unlike terms

3p + 2q

cannot be written as

5pq

or

5p

. Only terms with the same letters to the same powers combine.

Indices and Square Roots

Learn this subtopic in the notes

The laws of indices

Laws of indices

a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn} \qquad a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}

$a$ the base, not zero
$m, n$ the indices (powers)

Square roots of perfect squares

Square of a number ending in 5

(10n + 5)^2 = 100\,n(n + 1) + 25

$n$ the tens part of the number

Nested square roots

Infinite nested roots

\sqrt{n(n - 1) + \sqrt{n(n - 1) + \cdots}} = n \qquad \sqrt{n(n - 1) - \sqrt{n(n - 1) - \cdots}} = n - 1

Common traps

Adding indices for different bases

2^3 \times 3^2

is

8 \times 9 = 72

, not

6^5

. The add-the-indices law works only when the bases are the same.

Multiplying indices in a product

a^3 \times a^4 = a^7

, not

a^{12}

. Indices multiply only for a power of a power:

(a^3)^4 = a^{12}

.

Reading a zero index as zero

7^0 = 1

, not 0. Any non-zero number to the power 0 is 1.

Trusting the last digit alone

A square ending in 6 can have a root ending in 4 or 6:

\sqrt{1296} = 36

but

\sqrt{1156} = 34

. Compare with

35^2 = 1225

to choose.

Taking the root of each term

\sqrt{64 + 36}

is

\sqrt{100} = 10

, not

8 + 6 = 14

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

Keeping the negative root

x^2 - x - 20 = 0

gives

x = 5

or

x = -4

. A square root is never negative, so

\sqrt{20 + \sqrt{20 + \cdots}} = 5

.

Using the shortcut on the wrong number

\sqrt{10 + \sqrt{10 + \cdots}}

is not a whole number, because 10 is not a product of two consecutive numbers. Solving

x^2 - x - 10 = 0

gives

x = \frac{1 + \sqrt{41}}{2}

, about 3.7.

Simplification Shortcuts

Learn this subtopic in the notes

Difference of squares and cubes as a shortcut

Square and cube identities

a^2 - b^2 = (a + b)(a - b) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2) \qquad a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad (a + b)^2 = a^2 + 2ab + b^2

Sums of cubes in a fixed ratio

\frac{a^3 + b^3}{(ka)^3 + (kb)^3} = \frac{1}{k^3}

$k$ the multiple linking each term below to its partner above

Common factors and quick answer checks

Common factor

ab + ac = a(b + c) \qquad ab - ac = a(b - c)

Common traps

Pairing a cube with the wrong middle sign

\frac{a^3 - b^3}{a^2 + ab + b^2} = a - b

, but

\frac{a^3 - b^3}{a^2 - ab + b^2}

does not simplify. The middle term is plus for

a^3 - b^3

and minus for

a^3 + b^3

; check it before using the shortcut.

Dropping the middle term of a square

(30 + 0.5)^2

is

900 + 30 + 0.25 = 930.25

, not

900.25

. The term

2ab

is the one people forget.

Using k instead of k cubed

If each term below is 5 times its partner above, the bottom is

5^3 = 125

times the top, so the value is

\frac{1}{125} = 0.008

, not

\frac{1}{5} = 0.2

.

Pairing terms by position

Match each term with the one it is a multiple of, not the one in the same place. In

\frac{0.6^3 + 0.2^3}{0.1^3 + 0.3^3}

the 0.6 pairs with 0.3 and the 0.2 with 0.1:

k = 2

both times, so the value is 8.

Treating a passed check as proof

Two options can share a last digit or a digit sum. A check removes the options that fail it; it confirms the one left only when it is the only one.

Losing the sign in a remainder check

A subtraction stays a subtraction: if the parts leave 1 and 4, their difference leaves

1 - 4 = -3

, which is the same as a remainder of 6 on division by 9.

Surds and Rationalisation

Learn this subtopic in the notes

Simplifying, adding and expanding surds

Surd rules

\sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \qquad (\sqrt{a} + \sqrt{b})^2 - (\sqrt{a} - \sqrt{b})^2 = 4\sqrt{ab}

Rationalising with the conjugate

Conjugate

(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b \qquad \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b}

Comparing sums and differences of surds

Comparing surd sums and differences

(\sqrt{a} + \sqrt{b})^2 = (a + b) + 2\sqrt{ab} \qquad \sqrt{a} - \sqrt{b} = \frac{a - b}{\sqrt{a} + \sqrt{b}}

Common traps

Adding the numbers under the roots

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

, but

\sqrt{25} = 5

. Roots add only when they are like surds, and then you add the numbers in front of them.

Dropping the middle term of a surd square

(\sqrt{7} + 2)^2 = 7 + 4 + 4\sqrt{7} = 11 + 4\sqrt{7}

, not

7 + 4 = 11

. The middle term

2ab

carries the surd.

Forgetting to divide by a minus b

\frac{1}{\sqrt{7} - \sqrt{2}} = \frac{\sqrt{7} + \sqrt{2}}{5}

, not

\sqrt{7} + \sqrt{2}

. The reciprocal is the bare conjugate only when the two squares differ by 1.

Missing a negative denominator

\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} - \sqrt{5}}

is negative, because

\sqrt{3} < \sqrt{5}

. It equals

\frac{8 + 2\sqrt{15}}{-2} = -(4 + \sqrt{15})

, and the same value without its minus sign is the wrong option SSC likes to print.

Calling two surd sums equal because the numbers under the roots add up the same

\sqrt{7} + \sqrt{1}

and

\sqrt{4} + \sqrt{4}

both have a total of 8 under the roots, but they are not equal: about 3.646 against 4. Equal totals say nothing; compare the products, 7 against 16.

Squaring when a side may be negative

Squaring keeps the order of positive numbers only:

-3 < 2

but

9 > 4

. Check that both sides are positive before squaring them.

Square Roots of Surds and Surd Equations

Learn this subtopic in the notes

The square root of a + 2√b

Square root of a surd

\sqrt{a \pm 2\sqrt{b}} = \sqrt{m} \pm \sqrt{n} \quad \text{where } m + n = a,\ mn = b,\ m > n

Equations with a sum and a difference of square roots

Sum times difference of roots

(\sqrt{A} + \sqrt{B})(\sqrt{A} - \sqrt{B}) = A - B

Componendo and dividendo

\frac{p}{q} = \frac{a}{b} \;\Rightarrow\; \frac{p + q}{p - q} = \frac{a + b}{a - b} \qquad \frac{\sqrt{A} + \sqrt{B}}{\sqrt{A} - \sqrt{B}} = k \;\Rightarrow\; \frac{A}{B} = \left(\frac{k + 1}{k - 1}\right)^2

Common traps

Not writing the surd part as 2√b first

In

11 + 6\sqrt{2}

the surd part is

2\sqrt{18}

, so look for

m + n = 11

and

mn = 18

: 9 and 2, giving

3 + \sqrt{2}

. Using

mn = 2

or

mn = 6

finds no pair.

Putting the smaller root first after a minus

\sqrt{7 - 2\sqrt{12}}

is

2 - \sqrt{3}

, not

\sqrt{3} - 2

. A square root is never negative, so the larger root goes first.

Squaring a sum of roots term by term

\sqrt{x + 7} + \sqrt{x - 5} = 6

does not give

(x + 7) + (x - 5) = 36

. Squaring a sum adds the middle term

2\sqrt{(x + 7)(x - 5)}

, which is why the sum-and-difference route is quicker.

Forgetting to square the ratio of the roots

From

\frac{\sqrt{A}}{\sqrt{B}} = 3

the ratio

\frac{A}{B}

is 9, not 3. Componendo and dividendo gives the ratio of the roots; square it before solving for

x

.

Surds, Indices and Simplification formulas

BODMAS and Brackets

The order of operations (BODMAS)

Removing nested brackets from the inside out

The word 'of' and the bar in BODMAS

Common traps

Fractions and Decimals

Fractions and mixed numbers in BODMAS

Working with decimals

Comparing and ordering fractions

Continued fractions

Common traps

Missing Numbers and Letters

Simplifying and evaluating expressions in letters

Common traps

Indices and Square Roots

The laws of indices

Square roots of perfect squares

Nested square roots

Common traps

Simplification Shortcuts

Difference of squares and cubes as a shortcut

Common factors and quick answer checks

Common traps

Surds and Rationalisation

Simplifying, adding and expanding surds

Rationalising with the conjugate

Comparing sums and differences of surds

Common traps

Square Roots of Surds and Surd Equations

The square root of a + 2√b

Equations with a sum and a difference of square roots

Common traps

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