Exponents and Powers

12.1 INTRODUCTION
Exponent
If a certain number a is multiplied m times in succession, then the continued product so obtained is called the mth power of a and is written as am (read as, a to the power m).
Thus, am = a × a × a × a…….. to m factors.
Here, a is called the base of am and m is called the index or exponent of am.
Examples
i) x5=x.x.x.x.x
ii) (–3)6 = (–3)(–3)(–3)(–3)(–3)(–3)
Note
In particular, a2 is called the square of a (or, a2) and a3 is called the cube of a (or a3)
Root
If a and x are two real numbers and n is a positive integer such that an = x, then a is called the nth root of x and is written as a=xn or a=x1/n
Clearly, nth root of x (i.e., xn) is such a number whose nth power is equal to x i.e., (xn)n=x.
i) In particular, if a2=x, then a is called the second root or square root of x and is written as a=x2 or a=x1/2  or simply a=x
ii) If a3 = x, then a is called the third root or cube root of x and is written as a=x3 or a=x1/3
Examples
i) Square Root of 25 is 5 i.e., 25=5 52=25
ii) Cube Root of 27 is 3 i.e., 3 273=3 33=27
iii) Sixth Root of 64 is 2 i.e., 6 646=2 26=64
Example
Since 52=25 25=5
Again, 52=25 25=5
Therefore, it is evident that both 5 and (–5) are square roots of 25.
Hence, by the Square root of a real positive number x we mean ±x( i.e., +x or x)
Note
i) If x > 0 and n is any positive integer, then xn is positive.
ii) If x < 0 and n is any odd integer, then xn is negative.
iii) If x < 0 and n is any positive even integer, then xn does not exist in the set of real numbers.
12.2. LAWS OF EXPONENTS
I) Multiplication property :
am×an=am+nm, nz+ (Fundamental Index Law)
For multiplying the power of same base, powers are added.
Proof :

am×an=(a×a×a×. to m factors)

×(a×a×a×a . to n factors)

=(a×a×a×a.. to (m+n) facors)

 am×an=am+n

Examples :

i 23×24=23+4am×an=am+n

 23×24=27

ii (x+y)2×(x+y)3=(x+y)2+3

am×an=am+n

 (x+y)2×(x+y)3=(x+y)5

II) Division property property :

am÷an=amnm,nz+

For dividing the powers of same base, we subtract the indices.

Proof :

am÷an=aman=a×a×a×.. to m factors a×a×a×.. to n factors 

Case – 1 : If m>n

aman=a×a×a×a(mn) factors 

aman=amn

Example : 35÷32=3532=352=33aman=amn

Case –2 : If m<n

aman=1a×a×a×(nm) factors 

aman=1a(nm)

Example : 32÷35=3235=3×33×3×3×3×3=13×3×3

32÷35=133

III) Power of a power property :

amn=amnm,nz+

Proof : amn=am×am×am× to n factors =am+m+m+ to n factors 

amn=amn

Examples

i) 532=53×2=56  amn=amn

ii) y24=y2×4=y8

y24=y8

IV) Power of a product property :

am·bm=(ab)m m,nz+

Proof :

am×am=(a×a×a×.. to m factors)×(b×b×b×.. to m factors)

=(ab)×(ab)×(ab)× to m facors

 am×bm=(ab)m

Example : x3·y3=(x×x×x)×(y×y×y)

=(xy)×(xy)×(xy)=(xy)3

 x3.y3=(xy)3

V) Power of a divison property :

ambm=abm m,nz+

Proof : ambm=a×a×a× to m factors b×a×b× to m factors =ab×ab×ab×.. to m factors

ambm=abm

all the above laws are defind for m, nz+ only.
12.3. NUMBERS WITH NON-INTEGER EXPONENTS
Numbers with non-integer exponents
What if m is not a positive integer ?
If m is not a positive integer, then there exist four cases.
Case-1 : either m = 0
Meaning of a° (a 0)
We know the Fundamental Index law being true for all indices hence,

a°·am=a+m=am

Now Dividing both sides by am we get,

a°=amam=1(a0)

Note : a° has no meaning when a = 0 i.e., 0° has no meaning.
Case-2 : Either m=pq where p and q are positive integers.
Meaning of ap/q, where p and q are positive integer
Since q is a positive integer, hence,

ap/qq=ap/q·ap/q·ap/q.. to q factors 

=ap/q+p/q+p/q to q factors =ap/q·q=ap

 ap/q=apq

Thus, ap/q is the qth root of ap.
Case-3 : Either m is a negative number
Meaning of am , where m is a positive real number and a0
Since the fundamental index law is true for all indices, hence,

am·am=am+m=a°=1

am=1am and 1am=am

Thus am is the reciprocal of am.

Case-4 : Either m=pq where, p and q are positive integers.
We have, ap/q=1ap/q=1apq

Equations and identities involving Indices

If a, m, n are three real numbers and am=an, then m=n(a0,1,)
Proof : Since am=an and a0,1, hence, aman=1 or amn=a°

mn=0m=n proved

12.4. RATIONAL EXPONENT

I) Rational Number

The fraction pqwhere p and q are integers, q0 is called a rational number.

II) Positive Rational Exponent

Let a be a positive real number and n a positve fraction equal to pq, i.e., n=pq where p and q are positive integers, the equation xq=ap has one and only positive solution for x, given by x=apq=apq=qth root of ap.

III) Negative Rational Exponent

If n is a negative rational, i.e. n=pq, where q0 and a is a positive real number, then an=apq=1apq

Consider, xpq=1xpqxm=1xm=1xpq

xpq is the reciprocal of xpq

Thus, if x=rsr,s>0, then rspq=srpq

rsm=srm where mQ

IV) Also, all the laws of indices applicable for integral index are also applicable for rational index.
i.e., if a is a positive real number and m, n are rational numbers, then,

i) am×an=am+n

ii) amn=amn

iii) (ab)m=ambm

12.5. RADICAL AND RADICAND

We know that if y>0 and y1q=x, then we can write it as x= yq

Here, yq y is called ‘Radical form’ of y1q

i) In yq,  is called Radical Sign
ii) yq is called a Radical
iii) q is called Index of the radical
iv) y is called the Radicand
Note : Index of a radical is always positive
Important Results
i) ann=a
ii) abn=anbn
iii) anbn=abn
iv) anm=amn
v) amn=amn

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