OPERATIONS WITH SCIENTIFIC NOTATION

Operations with Scientific Notation :

In this section, you will learn 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.

(4 x 10⁵) + (0.1 x 10⁷)

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.

(0.723 x 10⁸) + (338.2 x 10⁵) - (6.1 x 10⁷)

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 = 10^14-8

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.

After having gone through the stuff above, we hope that the students would have understood how to do mathematical operations like addition, multiplication with scientific notation.

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