Indices and Surds — AFCAT Numerical Ability Formulas and Traps

The Laws of Exponents

Product, quotient and power of a power

The three index laws

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

Zero and negative indices

Negative index

a^{-n} = \dfrac{1}{a^n}, \qquad \left(\dfrac ab\right)^{-n} = \left(\dfrac ba\right)^{n}

Simplifying fractions of letters with powers

Letter by letter

\dfrac{p^a q^b}{p^c q^d} = p^{a-c}\, q^{b-d}

Common traps

Add indices only on the same base

4^3 \times 5^3

is

(4 \times 5)^3 = 20^3

, not

20^6

. Indices add when the base is the same; when only the index is the same, the bases multiply and the index stays.

A power of a power multiplies

(7^2)^3 = 7^6

, but

7^2 \times 7^3 = 7^5

. Brackets mean multiply the indices; a product means add them.

A negative index is not a negative number

3^{-3} = \tfrac{1}{27}

, not

-27

and not

-9

. The minus sign in the index means take the reciprocal.

A zero index gives 1, not 0

7^0 = 1

. The same holds for any non-zero base, and

x^{n} \times x^{-n} = x^0 = 1

.

Minus a negative is plus

In

\dfrac{x^6}{x^{-3}}

the index is

6 - (-3) = 9

, not

3

. Write the subtraction out with brackets before you do it.

The coefficient takes the power too

\left(\dfrac{3m}{n}\right)^{-2} = \dfrac{n^2}{9m^2}

. Flipping the fraction is not enough: the 3 is squared along with

m

.

Fractional Powers, Roots as Powers, and Equations in Powers

Learn this subtopic in the notes

A root is a fractional power

Root as an index

\sqrt[n]{a} = a^{1/n}, \qquad \sqrt[m]{\sqrt[n]{a}} = a^{1/(mn)}

Powers like m/n and decimal powers

Fractional index

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

Fractional indices on letters

Quotient with fractional indices

\dfrac{x^{a/b}}{x^{c/d}} = x^{\frac ab - \frac cd}

Equations in powers and in square roots

Equal powers of one base

a^{x} = a^{k} \;\Rightarrow\; x = k \quad (a > 0,\ a \ne 1)

Common traps

A half power is a square root, not half

36^{1/2} = 6

, not

18

. The index

\tfrac 12

asks for the number that squares to 36.

Nested roots multiply their indices

\sqrt[3]{\sqrt{x}} = x^{1/6}

, not

x^{1/5}

. A root of a root is a power of a power, so the indices multiply.

A decimal index is a fraction, not a divisor

x^{0.25}

is

x^{1/4}

, the fourth root of

x

. It is not

x

divided by 4, and

625^{0.25} = 5

, not

156.25

.

Same index: multiply bases. Different indices: one base

8^{0.2} \times 4^{0.2} = 32^{0.2} = 2

, because the indices match. With

8^{0.2} \times 4^{0.3}

they do not, so rewrite as

2^{0.6} \times 2^{0.6} = 2^{1.2}

.

Root first keeps the numbers small

1000^{2/3}

: the cube root is 10 and its square is 100. Squaring first means a cube root of a million, which is the same answer the long way.

The root of a cube is a three-halves power

\sqrt{a^3} = a^{3/2}

, not

a^{3}/2

and not

a^{6}

. The root divides the index by 2.

Subtract fractions over one denominator

\tfrac 45 - \tfrac 13

is

\tfrac{12 - 5}{15} = \tfrac{7}{15}

. Subtracting tops and bottoms separately gives

\tfrac 32

, which is wrong.

Isolate the root before squaring

In

\sqrt{x + 1} + 2 = 5

, first get

\sqrt{x + 1} = 3

, so

x = 8

. Squaring the left side as it stands does not remove the root.

Match the bases before matching indices

4^x = 8

does not give

x = 2

. Write it as

2^{2x} = 2^3

, so

x = 1.5

.

Surds: Simplest Form, Adding, Multiplying and Dividing

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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}

What a surd is: order, pure and mixed, like and unlike

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.

Common traps

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.

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.

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.

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.

Conjugates, Rationalising, and the Square Root of a Surd

Learn this subtopic in the notes

Squares and conjugate products of surds

Conjugate product

(\sqrt p + \sqrt q)(\sqrt p - \sqrt q) = p - q

Rationalising a denominator

Multiply by the conjugate

\dfrac{1}{a + \sqrt b} = \dfrac{a - \sqrt b}{a^2 - b}

The square root of a + 2√b

Square root of a surd expression

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

Common traps

A square keeps its middle term

(1 + \sqrt 3)^2 = 1 + 2\sqrt 3 + 3 = 4 + 2\sqrt 3

, not

1 + 3 = 4

. Squaring each term alone drops the

2ab

.

A conjugate product has no root left

(\sqrt 5 + 1)(\sqrt 5 - 1) = 5 - 1 = 4

. If a root is still in your answer, a cross term was not cancelled.

Square the surd, not the number inside

(4 + \sqrt 3)(4 - \sqrt 3) = 16 - 3 = 13

, because

(\sqrt 3)^2 = 3

. Writing

16 - 9

squares the 3 a second time.

Multiply the top as well

Multiplying only the bottom by the conjugate changes the value of the fraction. Top and bottom both take the same factor.

Change only the middle sign

The conjugate of

3 - \sqrt 7

is

3 + \sqrt 7

. Changing the sign of the 3 instead gives

(3 - \sqrt 7)(-3 + \sqrt 7) = -16 + 6\sqrt 7

, and the root stays in the bottom.

Make it 2√b before you look for the pair

In

\sqrt{11 + 6\sqrt 2}

, the pair does not multiply to 2. Write

6\sqrt 2 = 2\sqrt{18}

: the pair is 9 and 2, and the root is

3 + \sqrt 2

.

With a minus sign, the larger root goes first

\sqrt{7 - 2\sqrt{10}} = \sqrt 5 - \sqrt 2

. The order

\sqrt 2 - \sqrt 5

is negative, and a square root never is.

The root of a square is a modulus

\sqrt{(x - 4)^2} = |x - 4|

, not

x - 4

. The two agree only when

x \ge 4

.

Indices and Surds formulas

The Laws of Exponents

Product, quotient and power of a power

Zero and negative indices

Simplifying fractions of letters with powers

Common traps

Fractional Powers, Roots as Powers, and Equations in Powers

A root is a fractional power

Powers like m/n and decimal powers

Fractional indices on letters

Equations in powers and in square roots

Common traps

Surds: Simplest Form, Adding, Multiplying and Dividing

Simplest form: take out the largest square

Adding and subtracting like surds

Multiplying and dividing surds

What a surd is: order, pure and mixed, like and unlike

Common traps

Conjugates, Rationalising, and the Square Root of a Surd

Squares and conjugate products of surds

Rationalising a denominator

The square root of a + 2√b

Common traps

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