# Dividing Numbers in Scientific Notation

Here, we will learn how to divide numbers in scientific notation, which are written in the form (a x 10n)   where 1 ≤ a < 10. The number ‘a’ is the coefficient, and ‘b’ is the power or exponent.

## Steps with Examples

1. Separate and divide the coefficient and exponents separately.
2. Divide the bases with the help of the division rule of exponents (am ÷ an = am – n), and thus the exponents of the denominator are subtracted from the numerator.
3. Join the result of coefficients by the new power of 10
4. If the quotient from the division of coefficients is not within the range 1 ≤ a < 10, convert it to scientific notation form and then multiply it by the new power of 10

With Positive Exponents

Example – 1:

Divide and express in scientific notation: (8 x 108) ÷ (2 x 105)

1. Separating the coefficient and the exponential part

= (8 ÷ 2) × (108 ÷ 105)

• Dividing the coefficient and the exponential part separately

= 8 ÷ 2 = 4, 108 ÷ 105 = 108-5 = 103

• The coefficient is within the range 1 ≤ a < 10, thus multiplying it by the new power of 10.

Thus, the answer is 4 × 103

Example – 2:

Having Negative Exponent

Divide and express in scientific notation: (2 x 104) ÷ (4 x 10-7)

1. Separating the coefficient and the exponential part

= (2 ÷ 4) × (104 ÷ 10-7)

• Dividing the coefficient and the exponential part separately

= 2 ÷ 4 = 0.5, 104 ÷ 10-8 = 104-(-7) = 1011

• The coefficient is not within the range 1 ≤ a < 10 and is less than 1, thus converting it to scientific notation

= 0.5 = 5 × 10-1

• Now, multiplying the coefficient by the new power of 10, we get

= (5 × 10-1) × 1011

= 5 × 1010

Thus, the answer is 5 × 1010

Let us solve some more examples.

Divide 4.2 × 104 by 2.9 × 102 and express your answer in scientific notation.

Solution:

Separating the coefficient and the exponential part
= (4.2 ÷ 2.9) × (104 ÷ 102)
Dividing the coefficient and the exponential part separately
= 4.2 ÷ 2.9 = 1.448, 104 ÷ 102 = 104-2 = 102
The coefficient is within the range 1 ≤ a < 10, thus multiplying it by the new power of 10.
Thus, the answer is 1.448 ×102

Divide 3.2 × 104 by 5.7 × 10-2 and express your answer in scientific notation.

Solution:

Separating the coefficient and the exponential part
= (3.2 ÷ 5.7) × (104 ÷ 10-2)
Dividing the coefficient and the exponential part separately
= 3.2 ÷ 5.7 = 0.561, 104 ÷ 10-2 = 104-(-2) = 106
The coefficient is not within the range 1 ≤ a < 10 and is less than 1, thus converting it to scientific notation
= 0.561
= 5.61 × 10-1
Now, multiplying the coefficient by the new power of 10, we get
= (5.61 × 10-1) × 106
= 5 × 105
Thus, the answer is 5.61 × 105