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6. Simplify:

3
(a) 3 × 10 3 (b) 3 × 4 2 (c) 7 × 5 1 (d) 0 × 2 4
2
5
(e) 2 × 3 3 (f) (–3) × (–2) 3 (g) (–3) × (–5) 2 (h) (–2) × (–10) 4
3
3
4
Laws of Exponents
There are certain rules or laws that are applied to express mathematical expressions using
exponential notation.

These laws are called laws of exponents or exponential laws.
Let us discuss these laws one by one.
Law I: Multiplying Powers with the Same Base

Let us take few examples.
4
3
(a) 2 × 2 = (2 × 2 × 2) × (2 × 2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 7
Clearly, 2 × 2 = 2 = 2 3 + 4
7
4
3
2
3
(b) (−3) × (−3) = {(−3) × (−3)} × {(−3) × (−3) × (−3)}
= (−3) × (−3) × (−3) × (−3) × (−3) = (−3) 5
= (−3) 2 + 3
5
3
2
Clearly, (−3) × (−3) = (−3) = (−3) 2 + 3
Let a be a non-zero integer, then,
4
2
a × a = (a × a) × (a × a × a × a)
= a × a × a × a × a × a = a = a 2 + 4
6
4
Therefore, a × a = a 2 + 4 = a 6
2
It is clear from the above examples that if we multiply two quantities having the same base, then
the product is obtained by adding their powers or exponents, whereas the base remains the same.

n
In general if a be a non-zero integer and m, n are two whole numbers, then a × a = a m + n .
m
It is called the I law of exponents. It is also called the fundamental law of exponents.

n
m
p
Note: We can generalised the above law as a × a × a × … = a m + n + p + ... .
Example 9: Simplify the following in the exponential form:
(a) 3 × 3 2 (b) (–2) × (–2) 5 (c) x × x 2 (d) (–1) × (–1) 49
51
3
3
5
(e) 11 × 11 × 11 7
2
3
n
m
Solution: (a) 3 × 3 = 3 5 + 2 = 3 7 (Q a × a = a m + n )
5
2
n
m
5
3
(b) (–2) × (–2) = (–2) 3 + 5 = (–2) 8 (Q a × a = a m + n )
n
5
m
2
3
(c) x × x = x 3 + 2 = x (Q a × a = a m + n )
n
49
m
51
(d) (–1) × (–1) = (–1) 51 + 49 = (–1) 100 (Q a × a = a m + n )
p
2
m
3
n
7
(e) 11 × 11 × 11 = 11 3 + 2 + 7 = 11 12 (Q a × a × a = a m + n + p )
83 Exponents and Powers