OPERATIONS WITH SCIENTIFIC NOTATION
About "Operations with scientific notation"
Operations with scientific notation :
In this section, we are going to see, how to add, subtract, multiply, and divide numbers which are in scientific notation.
Operations with scientific notation - Examples
Example 1 :
Simplify the expression given below.
(4x10⁵) + (0.1x10⁷)
Solution :
Method 1 :
Step 1 :
In the given numbers, the highest power of 10 is 7.
So, write each number with 10 power 7.
4 x 10⁵ = 0.04 x 10⁷
0.1 x 10⁷ = 0.1 x 10⁷
Step 2 :
Add the multipliers for each number.
0.04 + 0.1 = 0.14
Step 3 :
Write the final answer in scientific notation :
0.14 x 10⁷ = 1.4 x 10⁶
Method 2 :
Step 1 :
First, write each number in standard notation.
4 x 10⁵ = 400,000
0.1 x 10⁷ = 1,000,000
Step 2 :
Find the sum of the numbers in standard notation.
400,000 + 1,000,000 = 1,400,000
Step 3 :
Write the final answer in scientific notation :
1,400,000 = 1.4 x 10⁶
Example 2 :
Simplify the expression given below.
(0.123x10⁸) + (338.2x10⁵) - (7.1x10⁷)
Solution :
Method 1 :
Step 1 :
In the given numbers, the highest power of 10 is 8.
So, write each number with 10 power 8.
0.723 x 10⁸ = 0.723 x 10⁸
338.2 x 10⁵ = 0.3382 x 10⁸
6.1 x 10⁷ = 0.61 x 10⁸
Step 2 :
Simplify the multipliers.
0.723 + 0.3382 - 0.61 = 0.4512
Step 3 :
Write the final answer in scientific notation :
0.4512 x 10⁸ = 4.512 x 10⁷
Method 2 :
Step 1 :
First, write each number in standard notation.
0.723 x 10⁸ = 72,300,000
338.2 x 10⁵ = 33,820,000
6.1 x 10⁷ = 61,000,000
Step 2 :
Simplify the numbers in standard notation.
72,300,000 + 33,820,000 - 61,000,000 = 45,120,000
Step 3 :
Write the final answer in scientific notation :
45,120,000 = 4.512 x 10⁷
Example 3 :
Multiply :
(3.2 x 10⁵) x (26.7 x 10³)
Solution :
Step 1 :
Find the product of multipliers for each number.
3.2 x 26.7 = 85.44
Step 2 :
Find the product of powers of 10.
10⁵ x 10³ = 10⁸
Step 3 :
Combine the results of step 1 and step 2 to write the final answer in scientific notation.
85.44 x 10⁸ = 8.544 x 10⁹
Example 4 :
When the Sun makes an orbit around the center of the Milky Way, it travels 2.025 × 10¹⁴ kilometers. The orbit takes 225 million years. At what rate does the Sun travel? Write your answer in scientific notation.
Solution :
Key points :
The answer is the number of kilometers per year that the Sun travels around the Milky Way.
Set up a division problem using
Rate = Distance / Time
to represent the situation.
Step 1 :
Substitute the values from the problem into the Rate formula.
Step 2 :
Write the expression for rate with years in scientific notation.
That is, 225 million = 2.25 x 10⁸
Then, we have
Step 3 :
Find the quotient by dividing the decimals and using the laws of exponents.
Divide the multipliers.
2.025 ÷ 2.25 = 0.9
Divide the powers of 10.
10¹⁴ ÷ 10⁸ = 10¹⁴ ⁻ ⁸
10¹⁴ ÷ 10⁸ = 10⁶
Step 4 :
Combine the answers to write the rate in scientific notation.
0.9 x 10⁶ = 9.0 x 10⁵
Justify and Evaluate :
Use estimation to check the reasonableness of your answer.
9.0 x 10⁵ is close 10⁶, so the answer is reasonable.
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