Power of 10

What is the Power of Ten?

The power of ten is ten multiplied by itself a certain number of times. It is a way of writing very large and very small sizes compactly.

By the power of 10, we mean 10 multiplying itself several times.

1. Examples of Power of 10, greater than or equal to 1 ( 10⁰ ≥ 1)

10⁰   =  1

10¹  =  10

10²  =  10 × 10  =  100

10³  = 10 × 10 × 10  =  1000

10⁴  =   10 × 10 × 10 × 10  =   10,000

10⁵   =  10 × 10 × 10 × 10 × 10   =  100,000

10⁶   =  10 × 10 × 10 × 10 × 10 × 10  =  1,000,000

2. Example of Power of 10 less than 1  (10 < 1)

Power of 10 is less than 1 if the index is a negative number. In this case, the number of zero(s) is after the decimal point  e.g

10^-1  = \( \frac{1}{10} \scriptsize = 0.1 \)

10^-2  = \( \frac{1}{10^2} \scriptsize = 0.01 \)

10^-3  = \( \frac{1}{10^3} \scriptsize = 0.001 \)

10^-4  = \( \frac{1}{10^4} \scriptsize = 0.0001 \)

10^-5  = \( \frac{1}{10^5} \scriptsize = 0.00001 \)

10^-6   = \( \frac{1}{10^6} \scriptsize = 0.000001 \)

Worked Example 6.1.1:

Convert these powers of 10 to figures:

a. 10⁴

b. 10⁵

c. 10^-5

Solution

a. 10⁴ = 10 × 10 × 10 × 10 = 10 000

b. 10⁵ = 10 × 10 × 10 × 10 × 10 =  100  000

c. 10^-5  = \( \frac{1}{10^5} \scriptsize = 0.00001 \)

Worked Example 6.1.2:

Express these as powers of 10:

a. 10 × 10 × 10 × 10 × 10
b. 10 × 10 × 10
c. One million
d. One-millionth
e. One billionth
f. One hundred thousand

Solution

a. 10 × 10 × 10 × 10 × 10  =  10⁵

b. 10 × 10 × 10   =  10³

c. One million = 1,000,000 =  10⁶

d. One millionth = \( \frac{1}{1000000} \\ = \frac{1}{10^6} \\ = \scriptsize 10^{-6} \)

e. One billionth = \( \frac{1}{1000000000} \\= \frac{1}{10^9}\\ = \scriptsize 10^{-9} \)

f. One hundred thousand   =  100,000  =  10⁵