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Solution:

Constants:
1 1
(a) 8 (b) –9 (f) (h) 0 (j) –
2 2
Variables:
(c) z (d) x (e) a (g) x, y (i) l, m, n

Powers on Variables

We know that the repeated product of numbers with itself is written in the form of their powers
or in exponential form. As,

4
2 × 2 × 2 × 2 = 2 is multiplied by itself 4 times = 2
3 × 3 × 3 × 3 × 3 × 3 = 3 is multiplied by itself 6 times = 3 6

2 × 2 × 2 × 7 × 7 = 2 is multiplied by itself thrice × 7 is multiplied by itself twice
3
= 2 × 7 2
3 × 3 × 5 × 5 × 6 × 6 × 6 = 3 is multiplied by itself twice × 5 is multiplied by itself twice × 6 is multiplied
2
2
by itself thrice = 3 × 5 × 6 3
In the same way, the repeated product of variables with itself is written in the form of their powers
or in exponential form.
As, x × x = x is multiplied by itself twice = x (read as x square or x to the power 2)
2
3
x × x × x = x is multiplied by itself thrice = x (read as x cube or x to the power 3)
y × y × y × y × y × y = y is multiplied by itself 6 times = y 6
p × p × q × q = p is multiplied by itself twice × q is multiplied by itself twice
2
= p × q = p q
2
2 2
p × p × p × q × q × r × r × r = p is multiplied by itself thrice × q is multiplied by itself twice × r is
3
2
3
3
2 3
multiplied by itself thrice = p × q × r = p q r
Also, 7 × 7 × x × x × x = 7 is multiplied by itself 2 times × x is multiplied by itself 3 times
= 7 × x = 7 x
2
3
3
2
Clearly, the exponent is the power of variables or numbers indicates that the number of time
variables or numbers are multiplied by itself.
Example 7: Express the following in exponential form:
(a) x × x × x × x × x Remember

(b) y × y × y × y × y × … 10 times The x means x.
1
1
(c) a × a × b × b × b × c × c Therefore, x = x.
(d) p × p × q × q × r × r
(e) 2 × 2 × (x × x × x × … 7 times)
(f) 2 × 3 × 3 × (m × m × m × … 5 times) × (n × n × n × … 7 times)

(g) 3 × 3 × p × p × 5 × q × q × 7 × r × r × r

107 Introduction to Algebra