EXPRESS THE GIVEN EXPRESSION IN INDEX FORM

Example 1 :

Write each number in index form:

a) 2 x 2 x 3 x 3 x 3

b) 2 x 5 x 5

c) 2 x 3 x 3 x 3 x 5

d) 5 x 5 x 7 x 7

e) 2 x 2 x 5 x 5 x 5 x 7

f) 3 x 3 x 7 x 7 x 11 x 11

g) 2 x 2 x 2 x 3 x 3 x 3 x 3

h) 5 x 5 x 5 x 5 x 5 x 5 x 7 x 7 x 7

Solution :

a) 2 x 2 x 3 x 3 x 3

2 is repeating 2 times and 3 is repeating 3 times.

=  2² x 3³

b) 2 x 5 x 5

5 is repeating two times.

=  2x5²

c) 2 x 3 x 3 x 3 x 5

3 is repeating 3 times.

=  2x3³x5

d) 5 x 5 x 7 x 7

5 is repeating 2 times and 7 is repeating 2 times.

=  5²x7²

e) 2 x 2 x 5 x 5 x 5 x 7

2 is repeating 2 times and 5 is repeating 5 times.

=  2²x5³x7

f) 3 x 3 x 7 x 7 x 11 x 11

3 is repeating 2 times, 7 is repeating 2 times and 11 is repeating 2 times.

=  3²x7²x11²

g) 2 x 2 x 2 x 3 x 3 x 3 x 3

2 is repeating 3 times and 3 is repeating 4 times.

=  2³x3⁴

Example 2 :

Write the following in index form with 2 as a base:

a) 16     b) 4     c) 64

Solution :

(a)  16  =  2 x 2 x 2 x 2

Instead of writing 2 four times, we write

16  =  2⁴

(b)  4

4  =  2 x 2

Instead of writing 2 two times, we write

4  =  2²

(c)  64

64  =  2 x 2 x 2 x 2 x 2 x 2

Instead of writing 2 two times, we write

64  =  2⁶

Example 3 :

Write the following in exponent form with 10 as a base :

a) 100        b) 1000 

Solution :

a) 100 

100  =  10 x 10  =  10²

a) 1000 

1000  =  10 x 10 x 10  =  10³

Example 4 :

Express the numbers appearing in the following statements in the form k × 10ⁿ where 1 ≤ k < n and n is an integer.

(i) The mean distance of the moon from the earth is 384,400,000 metres.

(ii) The distance travelled by a ray of light in one year is 9,460,500,000,000,000 m.

(iii) The number of red blood cells per cubic mm of human blood is approximately 5.5 millions.

Solution :

i)  The mean distance between moon from earth

= 384,400,000 metres

= 3844 x 100000

= 3844 x 10⁵

(ii) The distance travelled by a ray of light in one year

= 9,460,500,000,000,000 m

= 94605 x 100,000,000,000

= 94605 x 10¹¹

(iii) The number of red blood cells per cubic mm of human blood

= approximately 5.5 millions.

1 million = 1000000

5.5 million = 5.5 x 1000000

= 55 x 100000

= 55 x 10⁵

Example 5 :

If (p/q) = (5/6)^-2 x (4/3)⁰, find the value of (p/q)^-2

Solution :

(p/q) = (5/6)^-2 x (4/3)⁰

The value of anything to the power 0 is 1.

(p/q) = (5/6)^-2 x 1

(p/q) = (5/6)^-2

Put Power -2 on both sides, we get

(p/q)^-2 = [(5/6)^-2]^-2

(p/q)^-2 = (5/6)^-2(-2)

(p/q)^-2 = (5/6)⁴

Example 6 :

If 2^{2x - 3} = (64)^x, find the value of x.

Solution :

Since we have 2 as a base in the left side, we will try to express 64 also with the base of 2.

2^{2x - 3} = (64)^x

64 = 2⁶

2^{2x - 3} = 2⁶

Equating the power, we get

2x - 3 = 6

Adding 3, we get

2x = 6 + 3

2x = 9

dividing by 2, we get

x = 9/2

Problem 7 :

By what number should (7)^-2 be multiplied so that the product may be equal to (343)^-1?

Solution :

Let the number to be multiplied as x.

(7)^-2 · x = (343)^-1

x = (343)^-1 /(7)^-2 

To do the more simplification, let us write 343 also with the base of 7.

Expressing 343 with the base of 7, we get

343 = 7³ 

x = (7³)^-1 /(7)^-2 

= 7^-3 /(7)^-2 

= 7^-3+2 

= 7^-1

Converting the negative exponent as positive, we get

= 1/7

So, the required quantity is 1/7.

Problem 8 :

9 × 4² is same as

(i) (12)² (ii) (36)² (iii) (36)³ (iv)  (18)⁴

Solution :

= 9 × 4²

By expressing 9 in exponential form, we get

= 3² × 4²

= (3 x 4)²

= 12²

So, option (i) is correct.

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