**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.

**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 10****⁶**

**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 10****⁶**

**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 10****⁹**

**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 10****⁵**

**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 given above, we hope that the students would have understood "Operations with scientific notation".

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