Table of Contents

Last modified on September 23rd, 2022

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

- Separate and divide the coefficient and exponents separately.
- Divide the bases with the help of the division rule of exponents (a
^{m}÷ a^{n}= a^{m – n}), and thus the exponents of the denominator are subtracted from the numerator. - Join the result of coefficients by the new power of 10
- 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 10 ^{8}) **

- Separating the coefficient and the exponential part

= (8 ÷ 2) × (10^{8} ÷ 10^{5})

- Dividing the coefficient and the exponential part separately

= 8 ÷ 2 = 4, 10^{8 }÷ 10^{5} = 10^{8-5} = 10^{3}

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

Thus, the answer is 4 × 10^{3}

**Example – 2:**

**Having Negative Exponent**

**Divide and express in scientific notation: (2 x 10 ^{4}) **

- Separating the coefficient and the exponential part

= (2 ÷ 4) × (10^{4} ÷ 10^{-7})

- Dividing the coefficient and the exponential part separately

= 2 ÷ 4 = 0.5, 10^{4 }÷ 10^{-8} = 10^{4-(-7)} = 10^{11}

- 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}) × 10^{11}

= 5 × 10^{10}

Thus, the answer is 5 × 10^{10}

Let us solve some more examples.

**Divide 4.2 × 10 ^{4} by 2.9 × 10^{2} and express your answer in scientific notation.**

Solution:

Separating the coefficient and the exponential part

= (4.2 ÷ 2.9) × (10^{4} ÷ 10^{2})

Dividing the coefficient and the exponential part separately

= 4.2 ÷ 2.9 = 1.448, 10^{4 }÷ 10^{2} = 10^{4-2} = 10^{2}

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

Thus, the answer is 1.448 ×10^{2}

**Divide 3.2 × 10 ^{4} by 5.7 × 10^{-2}**

Solution:

Separating the coefficient and the exponential part

= (3.2 ÷ 5.7) × (10^{4} ÷ 10^{-2})

Dividing the coefficient and the exponential part separately

= 3.2 ÷ 5.7 = 0.561, 10^{4 }÷ 10^{-2} = 10^{4-(-2)} = 10^{6}

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}) × 10^{6}

= 5 × 10^{5}

Thus, the answer is 5.61 × 10^{5}