CAT Quant: Indices, Surds and Logarithms – Important Formulas and Concepts

Indices – Surds

Indices, also known as exponents or powers, are mathematical notations used to represent repeated multiplication of a number by itself.
If we multiply a number “a” by itself “n times” times, we express it as it as “aⁿ” (read as a raised to the power n). We say that ‘a’ is expressed as an exponent. Here, ‘a’ is called the ‘base’ and ‘n’ is called the ‘power / index / exponent’. $$a^n = a \times a \times a \times ……. \times (n \ times)$$

$$a^m \times a^n = a^{m+n} $$
$$\frac{a^m}{a^n} = a^{am-n}$$
$$(a^m)^n = a^{mn}$$
$$a^{-m} = \frac{1}{a^m}$$
$$a^m \times b^m = {(ab)}^m$$
$$\frac{a^m}{b^m} = {\LARGE(} \frac{a}{b} {\LARGE)} ^m$$
$${\LARGE(} \frac{a}{b} {\LARGE)} ^{-m} = {\LARGE(} \frac{b}{a} {\LARGE)} ^m$$
$$a^0 = 1 \\ [Except \ when \ a =0 \ (not \ defined)]$$
$$a^1 = a$$
$$1^n = 1 $$
$$(-1)^{even} = 1 \ ; \ (-1)^{odd} = -1$$
NOTE – If we have power in the form of steps $ {{{2}^a}^b}^c $ , then such a number is evaluated by starting at the topmost level of the step.

Surds are expressions that involve the square root of a number that cannot be simplified to rational numbers (i.e. it is not a perfect square). In other words, surds are irrational numbers expressed in radical form. For example $ \sqrt{2}, \sqrt{3}, (6 + \sqrt{5}) $.
Addition and subtraction are possible only when surds are like terms.

If there is a surd of the form $ (a+\sqrt{b}) $ then a surd of the form $ \pm(a-\sqrt{b}) $ is its conjugate.
For example, conjugate of $ (6 + \sqrt{2}) \ is \ (6 – \sqrt{2}) $.
Product of surd and conjugate will always be a rational number.
If a sud of the form $ \sqrt{a} + \sqrt{b} \sqrt{c} $ is in the denominator, the process of multiplying the denominator and numerator with its conjugate surd has to be done twice.

Rationalization of Surds –
- Rationalizing the denominator: Multiply the numerator and the denominator by a suitable surd to eliminate the root from the denominator.
- Surds when multiplied to its conjugate or its rationalizing factor (R.F) converts it into a rational number.
- For Example ; $ \sqrt{a} \times \sqrt{a} $ [R.F] = $ a $ $ \hspace{1cm} (\sqrt{a} + \sqrt{b}) \times (\sqrt{a} – \sqrt{b}) $ [R.F] = $ (a-b) $
Square Root of a Surd –
- In order to find out the square root of a surd the following method can be used.
- For the square root of a number of the form $ (a + b\sqrt{c}) $ Let the square root be $ \\ \longrightarrow \sqrt{a+b\sqrt{c}} = x = y\sqrt{c} $ On Squaring both sides we get $ \\ $ $ (a + b\sqrt{c}) = x^2 + {y}^{2}c + 2xy\sqrt{c} $ $ \\ $ where, [a and $ (x +y^{2}c) $ is rational] and [$ (b\sqrt{c}) and (2xy\sqrt{c}) $ is irrational] .
- Rational part on the left hand sid eis equal to rational part on right hand side and the irrational part on the left hand side is rqual to irrational part on the right side. These two equation are solved to arrive at the square root of a surd.

the logarithm of any number to a given base is the index or the power to which the base must be raised in order to equal the given number.
If $ a^x = N \ then \ x = log_{a} N $
This is read as “log n to the base a” where, N, a > 0 and a $ \ne $ 1.
For Example – 216 = 6³ can be expressed as $ log_{6} \ 216 = 3 $

Logarithms can be expressed to any base (positive number other than 0)Logarithms from one base can be converted to logarithms of any other base.
There are two types of logarithms that are commonly used, on the basis of bases –
- Natural (Napierian) Logarithms –Logarithms that are expressed to the base of a number called “e ” where “e” is Euler’s number approximately equal to 2.718 .
- Common Logarithms – Logarithms expressed to the base of “10”. If logarithms are given without mentioning any base, it can normally be taken to be logarithms to the base 10.

$$log_{a} a = 1$$ $$(Logarithm \ of \ any \ number \ to \ the \ same \ base \ is \ 1)$$
$$log_{a}1 = 0 $$ $$ (Logarithm \ of \ 1 \ to \ any \ base \ other \ than \ 1 \ is \ 0)$$
$$log_{a}(mn) = log_{a}m + log_{a}n$$
$$log_{a}{\frac{m}{n}} = log_{a}m – log_{a}n$$
$$log_{a}m^p = p \times log_{a}m$$
$$log_{a}b = \frac{1}{log_{b}a}$$
$$log_{a}m = \frac{log_{b}m}{log_{b}a}$$ $$(Change \ of \ Base)$$
$$log_{a^{q}}{m^p} = \frac{p}{q} \ log_{a}m$$
$$a^{log_{a}N} = N$$
$$a^{log {b}} = b^{log{a}} $$