NUMERIC PROCESS 1

INDICES

The lowest factors of 2000 are 2×2×2×2×5×5×5.
These factors are written as 24 ×53, where 2 and 5 are called bases and the numbers 4 and 5 are called indices. When an index is an integer it is called a power. Thus,2^4 is called ‘two to the power of four’, and has a base of 2 and an index of 4. Similarly, 5^3 is called ‘five to the power of  3’ and has a base of 5 and an index of 3. Special names may be used when the indices are 2 and 3, these being called ‘squared’and ‘cubed’, respectively. Thus 7^2 is called ‘seven squared’ and 9^3 is called ‘nine cubed’. When no index is shown, the power is 1, i.e. 2
means 2^1.

Reciprocal
The reciprocal of a number is when the index is −1 and its value is given by 1, divided by the base. Thus the reciprocal of 2 is 2^−1 and its value is 1/2 or 0.5. Similarly, the reciprocal of 5 is 5^−1 which means 1/5
or 0.2.


Square root
The square root of a number is when the index is 1^2 , and the square root of 2 is written as 2^1/2 or √2. The value of a square root is the value of the base which when multiplied by itself gives the number. Since 3×3=9,
then
√9=3. However, (−3)×(−3)=9, so
√9=−3.
There are always two answers when finding the square root of a number and this is shown by putting both a+and a−sign in front of the answer to a square
root problem. Thus
√9=}3 and 41/2 =
√4=}2,



Laws of indices
When simplifying calculations involving indices, certain basic rules or laws can be applied, called the laws of indices. These are given below.
(i) When multiplying two or more numbers having
the same base, the indices are added. Thus
3^2 ×3^4 =3^2+4 =3^6
(ii) When a number is divided by^ a number having the
same base, the indices are subtracted. Thus
3^5
3^2
=3^5−2 =3^3

(iii) When a number which is raised to a power is raised
to a further power, the indices are multiplied. Thus
(3^5)2 =3^5×2 =3^10
(iv) When a number has an index of 0, its value is 1.
Thus 3^0 =1
(v) A number raised to a negative power is the reciprocal
of that number raised to a positive power.
Thus 3^−4 = 1/3^4 Similarly, 1/2^−3 =2^3
(vi) When a number is raised to a fractional power the denominator of the fraction is the root of the number and the numerator is the power.
Thus 8^2/3 = 3√8^2 =(2)^2 =4
and 25^1/2 = 2√25^1 =
√25^1 =}5
(Note that
√≡ 2√)

 problems on indices
Problem 1. Evaluate (a) 5^2 ×5^3, (b) 3^2 ×3^4 ×3
and (c) 2×2^2 ×2^5
SOLUTION
From law (i):
(a) 5^2 ×5^3 =5^(2+3) =5^5 =5×5×5×5×5=3125
(b) 3^2 ×3^4 ×3=3^(2+4+1) =3^7
=3×3× ・ ・ ・ to 7 terms
=2187
(c) 2×2^2 ×2^5 =2^(1+2+5) =2^8 =256
Problem 2. Find the value of:
(a)
7^5/7^3 and (b)
5^7/5^4
SOLUTION
From law (ii):
(a)
7^5/7^3
=7^(5−3) =7^2 =49
(b)
5^7/5^4
=5^(7−4) =5^3 =125
Problem 3. Evaluate: (a) 5^2 × 5^3 ÷ 5^4 and
(b) (3×3^5)÷(3^2 ×3^3)
SOLUTION
From laws (i) and (ii):
(a) 5^2 ×5^3 ÷ 5^4 = 5^2 × 5^3
5^4
= 5^(2+3)/5^4
= 5^5/5^4
=5^(5−4) =5^1 =5
(b) (3×3^5)÷(3^2 ×3^3)= 3 × 3^5
3^2 × 3^3
= 3^(1+5)
= 3^6/3^5
= 3^(6−5) = 3^1 = 3
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