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
- Separate and divide the coefficient and exponents separately.
- 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.
- 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 108) ÷ (2 x 105)
- 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)
- 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.
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Divide 4.2 × 104 by 2.9 × 102 and express your answer in scientific notation.
Solution:
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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
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Divide 3.2 × 104 by 5.7 × 10-2 and express your answer in scientific notation.
Solution:
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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
