# 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 ${m}^{th}$ power of a and is written as ${a}^{m}$ (read as, a to the power m).
Thus, ${a}^{m}$ = a × a × a × a…….. to m factors.
Here, a is called the base of ${a}^{m}$ and m is called the index or exponent of ${a}^{m}$.
Examples
i) ${x}^{5}=x.x.x.x.x$
ii) (–3)6 = (–3)(–3)(–3)(–3)(–3)(–3)
Note
In particular, a2 is called the square of a (or, ${a}^{2}$) and a3 is called the cube of a (or ${a}^{3}$)
Root
If a and x are two real numbers and n is a positive integer such that an = x, then a is called the ${n}^{th}$ root of x and is written as
Clearly, ${n}^{th}$ root of x (i.e., $\sqrt[n]{x}$) is such a number whose ${n}^{th}$ power is equal to x i.e., $\left(\sqrt[n]{x}{\right)}^{n}=x$.
i) In particular, if ${a}^{2}=x$, then a is called the second root or square root of x and is written as
ii) If a3 = x, then a is called the third root or cube root of x and is written as
Examples
i) Square Root of 25 is 5 i.e.,
ii) Cube Root of 27 is 3 i.e., 3
iii) Sixth Root of 64 is 2 i.e., 6
Example
Since
Again,
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
Note
i) If x > 0 and n is any positive integer, then $\sqrt[n]{x}$ is positive.
ii) If x < 0 and n is any odd integer, then $\sqrt[n]{x}$ is negative.
iii) If x < 0 and n is any positive even integer, then $\sqrt[n]{x}$ does not exist in the set of real numbers.
12.2. LAWS OF EXPONENTS
I) Multiplication property :
(Fundamental Index Law)
For multiplying the power of same base, powers are added.
Proof :

Examples :

$\left[\because {a}^{m}×{a}^{n}={a}^{m+n}\right]$

II) Division property property :

${a}^{m}÷{a}^{n}={a}^{m–n}\left(m,n\in {z}^{+}\right)$

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

Proof :

Case – 1 : If $m>n$

$\therefore \frac{{a}^{m}}{{a}^{n}}={a}^{m–n}$

Example : ${3}^{5}÷{3}^{2}=\frac{{3}^{5}}{{3}^{2}}={3}^{5–2}={3}^{3}\left(\because \frac{{a}^{m}}{{a}^{n}}={a}^{m–n}\right)$

Case –2 : If $m

$\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{\left(n–m\right)}}$

Example : ${3}^{2}÷{3}^{5}=\frac{{3}^{2}}{{3}^{5}}=\frac{3×3}{3×3×3×3×3}=\frac{1}{3×3×3}$

${3}^{2}÷{3}^{5}=\frac{1}{{3}^{3}}$

III) Power of a power property :

${\left({a}^{m}\right)}^{n}={a}^{mn}\left(m,n\in {z}^{+}\right)$

Proof :

${\left({a}^{m}\right)}^{n}={a}^{mn}$

Examples

i)

ii) ${\left({y}^{2}\right)}^{4}={y}^{2×4}={y}^{8}$

$\therefore {\left({y}^{2}\right)}^{4}={y}^{8}$

IV) Power of a product property :

Proof :

Example : ${x}^{3}·{y}^{3}=\left(x×x×x\right)×\left(y×y×y\right)$

$=\left(xy\right)×\left(xy\right)×\left(xy\right)=\left(xy{\right)}^{3}$

V) Power of a divison property :

Proof :

$\therefore \frac{{a}^{m}}{{b}^{m}}={\left(\frac{a}{b}\right)}^{m}$

all the above laws are defind for 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 $\ne$ 0)
We know the Fundamental Index law being true for all indices hence,

$a°·{a}^{m}={a}^{\circ +m}={a}^{m}$

Now Dividing both sides by ${a}^{m}$ we get,

$a°=\frac{{a}^{m}}{{a}^{m}}=1\left(\because a\ne 0\right)$

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

Thus, ${a}^{p/q}$ is the ${q}^{th}$ root of ${a}^{p}$.
Case-3 : Either m is a negative number
Meaning of ${\mathrm{a}}^{–m}$ , where m is a positive real number and $a\ne 0$
Since the fundamental index law is true for all indices, hence,

${a}^{–m}·{a}^{m}={a}^{–m+m}=a°=1$

Thus ${a}^{–m}$ is the reciprocal of ${a}^{m}$.

Case-4 : Either $m=–\frac{p}{q}$ where, p and q are positive integers.
We have, ${a}^{–p/q}=\frac{1}{{a}^{p/q}}=\frac{1}{\sqrt[q]{{a}^{p}}}$

Equations and identities involving Indices

If a, m, n are three real numbers and
Proof : Since

12.4. RATIONAL EXPONENT

I) Rational Number

The fraction $\frac{p}{q}$where p and q are integers, $q\ne 0$ is called a rational number.

II) Positive Rational Exponent

Let a be a positive real number and n a positve fraction equal to where p and q are positive integers, the equation ${x}^{q}={a}^{p}$ has one and only positive solution for x, given by .

III) Negative Rational Exponent

If n is a negative rational, i.e. $n=–\frac{p}{q}$, where $q\ne 0$ and a is a positive real number, then ${a}^{n}={a}^{\frac{p}{q}}=\frac{1}{{a}^{\frac{p}{q}}}$

Consider, ${x}^{–\frac{p}{q}}=\frac{1}{{x}^{\frac{p}{q}}}\left[\because {x}^{–m}=\frac{1}{{x}^{m}}\right]={\left(\frac{1}{x}\right)}^{\frac{p}{q}}$

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) ${a}^{m}×{a}^{n}={a}^{m+n}$

ii) ${\left({a}^{m}\right)}^{n}={a}^{mn}$

iii) $\left(ab{\right)}^{m}={a}^{m}{b}^{m}$

We know that if then we can write it as

Here, $\sqrt[q]{\mathrm{y}}$ y is called ‘Radical form’ of ${y}^{\frac{1}{q}}$

i) In is called Radical Sign
ii) $\sqrt[q]{y}$ 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) $\sqrt[n]{{a}^{n}}=a$
ii) $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$
iii) $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
iv) $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$
v) $\sqrt[n]{{a}^{–m}}={a}^{–\frac{m}{n}}$