The document Power and Index Quant Notes | EduRev is a part of the Quant Course Quantitative Techniques for CLAT.

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**Chapter 2 Power and Index **

In p^{n}, p is called the base and n is called the power or index

In (-5)^{7}, -5 is te base and 7 is the power

In , is the base and 7 is the power

And

= **Remarks:** In the exponential notation, the base can be any rational number and the power can be any integer.**Note:** If the power of a rational number is 1 its value will be the rational number itself

i.e. (-10)^{1} = -10, and in general

.

**Example:** **Find the value of**

**Solution**: **Product law for exponents:** If p is a non-zero rational number and m and n are two positive integers then p^{m} x p^{n} = p ^{m+n}

also p^{m} x p^{n} x p^{r} x p^{s} = p^{m+n+r+s}

also (p^{m})^{n} = p^{mn}

**Quotient law for exponents:** It p is a non- zero rational number and m and n are two positive integers

then p^{m} Ã· p^{n} = p^{m-n} for m > n

and p^{m} Ã· p^{n} = for m < n

**If power is zero (o):** If p is a non-zero rational number then p^{o} = 1**If power is (-1):** If p is a non-zero rational number then p^{-1} denotes the reciprocal of p and (p)^{-1} = **A negative integer as power**

p^{-m} = **other laws of exponents**

p^{m} x q^{m} = (p x q)^{m}

**Few examples showing the application of laws of exponents.****Example1:** **Simplify (a) ****Solution:** a)

= 2^{-2} x 3^{-4+2} = 2^{-2} x 3^{-2}

**Example2:** **Find m if**

**Solution**

(a) LHS=

RHS =

Equating LHS and RHS

Because base is same, powers must be equal

So -2m + 1 = -27

or -2m = -27 â€“ 1

= -28

or m = 14

(b) LHS =

RHS = 2^{m}

So 2^{m} = 2^{5}

Or m = 5.

**Example:** **Solve for xa) 3 ^{x} = 81 b) (7^{2x})^{-2} = (2401)**

**Solution** a) RHS = 81 = 3^{4}

So 3^{x} = 3^{4} or x = 4**Solution** b) RHS= (2401)^{-1} = (7^{4})^{-1} = 7^{-4}

LHS = (7^{2x})^{-2} = 7^{-4x}

Equating LHS and RHS

7^{-4X} = 7^{-4}

Base in same, powers must be equal

-4x = -4

Or x = 1

**What is the difference between exponents**

= p^{m} x p^{m} x p^{m} x p^{m} --------- n times

=p^{m+m+m+m} ------ n times

=p^{m n}

Where as =

Let us simplify it with the help of an example

Find the difference between

(2^{2})^{3} = 2^{2} x 2^{2} x 2^{2} = 2^{6}

= 2^{2x2x2 }= 2^{8}

So the diff. is very clear.

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