We use the **cientific notation** to express very small numbers, such as 0.00000000003, or very large numbers, such as 123,500,000,000. Its representation is given in relation to base 10, that is:

**The. 10 ^{B}**

a = **coefficient**

10 = **base**

b = **exponent**

To represent a number in scientific notation, we must write it as a product. Look:

**2,3. 10 ^{+2}4,567. 10^{+3}394,56. 10^{-4}0,23563. 10^{-5}**

Note that base 10 exponents can be negative or positive. What determines this factor is the displacement of the comma. To understand this better, understand the following two rules:

**First Rule: **The base 10 exponent will be positive when we shift the comma to the left. In this case, we must add the number of displaced houses in the base 10 exponent. Watch.

**→ Number**: 23458

→ The comma is after the figure eight: 23458,

→ We will shift the comma four places to the left:

2345,8

234,58

23,458

2,3458

The comma has been shifted 4 places to the left, so we must multiply the number 2.3458 by a base 10 value, which will be: 10000 = 10^{+4}. Each zero of the 10000 represents one place in displacement. Since the shift has gone to the left, the base 10 exponent is positive.

**→ The number 23458 in scientific notation: 2.3458. 10000 = 2,3458. 10 ^{+4}**

**Second Rule: **The base 10 exponent will be negative when we shift the comma to the right. In this case, we must subtract the amount of shifted places in the base 10 exponent.

**→ Number**: 0,0014

→ We will shift the comma 3 places to the right:

00,014

000,14

0001,4

The comma has been shifted three places to the right, so we must multiply the number 1.4 by a base 10 value, which will be 1/1000 = 10^{-3}. Each zero represents a place in displacement. Since the shift has gone to the right, the base 10 exponent is negative.

→ The number 0.0014 written in scientific notation: 1.4. 10-3

**Examples**

**Write the numbers below as scientific notation.**

**The)** 0,00032

**B) **53000

**Solution:**

**The)** 0,00032

The comma must be shifted to the right. Thus, the base 10 exponent will be negative.

00,0032

000,032

0000,32

00003,2

As we shift the decimal point 4 places to the right, we represent scientific notation as follows: 3.2. 10^{-4}

**B) **53000

The comma must be shifted 4 places to the left. Thus, the base 10 exponent will be positive.

5300,0

530,00

53,000

5,3000

As we shift the decimal point 4 places to the left, we represent scientific notation as follows: 5.3. 10^{+4}.