**Exponents**

We know
that, 5 × 5 × 5 × 5 × 5 × 5 is written as 5

^{6}and is read as ‘5 raised to the power 6’. Here, 5 is called the base and 6 is called the exponent or power or index.
In general,
if a is a literal number and it is multiplied by itself, m number of times,
then a × a × a × ... × a (m times) = a

^{m}
This method
of expressing product of a number by itself in short form is called the
exponential notation or power notation.

**Example 1:**Express the following products in exponential form and evaluate.

a. 7 × 7 × 7
× 7 × 7 × 7 × 7 × 7

b. (–5) ×
(–5) × (–5) × (–5) × (–5)

**Solution:**

a. 7 × 7 × 7
× 7 × 7 × 7 × 7 × 7 = 7

^{8}= 5764801
b. (–5) ×
(–5) × (–5) × (–5) × (–5) = (–5)

^{5}= –3125**Laws of
Exponents **

If a and b
are any two real numbers, and m and n are two any integers, then

**Law 1:**a

^{m}× a

^{n}= a

^{m + n}

**Law 2:**a

^{m}÷ a

^{n}= a

^{m – n}, where a ≠ 0

**Law 3:**(a

^{m})

^{n}= a

^{m × n}

**Law 4:**a

^{m}× b

^{m}= (ab)

^{m}

**Law 5:**(a/b)

^{m}= a

^{m}/b

^{m}

**Law 6:**(a/b)

^{-}^{m}= (b/a)

^{m}, where a ≠ 0 and b ≠ 0

**Values
with Zero Exponents **

If a is any
real number, then a

^{0}= 1, where a ≠ 0.
For example,
12

^{0}= 1, (1245)^{0}= 1 and (235.121)^{0}= 1**Values
with Negative Exponents**

If a is any
real number, then a

^{–n}= 1/a^{n}and 1/a^{-n}= a^{n}where a ≠ 0.
For example,
5

^{-4}= 1/5^{4}= 1/625 and 1/4^{-3}= 4^{3}= 64**Value
with Fractional Exponents **

If a is any
real number,

^{n}√a = a^{1/n}and^{n}√a^{m}= a^{m/n}where a ≠ 0.
For example,

^{3}√8 = 8^{1/3}and^{5}√7^{3}= 7^{3/5}^{}

**Example 2:**Evaluate the following using laws of exponents.

a. p × p × p
× p × p × p × q × q × q × q × q × r

b. [(3

^{2})^{3}]^{4}
c. 2

^{0}× 5^{0}× 2^{–3}× 5^{–5}× 5^{3}
d. (4

^{2})^{3}× 4^{2}× 4^{–5}^{}

**Solution:**

a. p × p × p
× p × p × p × q × q × q × q × q × r = p

^{6}q^{5}r
b. [(3

^{2})^{3}]^{4}= [3^{2 × 3}]^{4}= 3^{6 × 4}= 3^{24}
c. 2

^{0}× 5^{0}× 2^{–3}× 5^{–5}× 5^{3}= 1 × 1 × 2^{–3}× 5^{–5 + 3}= 2^{–3}× 5^{–2}= 1/2^{3}× 1/5^{2}= 1/8 × 1/25 = 1/200
d. (4

^{2})^{3}× 4^{2}× 4^{–5}= 4^{2 × 3}× 4^{2}× 4^{–5}= 4^{(6 + 2 – 5)}= 4^{3}= 64