**Example 1 :**

Simplify the expression given below.

(4 x 10^{5}) + (0.1 x 10^{7})

**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 10^{5} = 0.04 x 10^{7}

0.1 x 10^{7} = 0.1 x 10^{7}

**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 10^{7} = 1.4 x 10^{6}

**Method 2 :**

**Step 1 : **

First, write each number in standard notation.

4 x 10^{5} = 400,000

0.1 x 10^{7 }= 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.

(0.723 x 10^{8}) + (338.2 x 10^{5}) - (6.1 x 10^{7})

**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 10^{8} = 0.723 x 10^{8}

338.2 x 10^{5} = 0.3382 x 10^{8}

6.1 x 10^{7 }= 0.61 x 10^{8}

**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 10^{8} = 4.512 x 10^{7}

**Method 2 :**

**Step 1 : **

First, write each number in standard notation.

0.723 x 10^{8} = 72,300,000

338.2 x 10^{5} = 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 10^{7}

**Example 3 :**

Multiply :

(3.2 x 10^{5}) x (26.7 x 10^{3})

**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.**

10^{5} x 10^{3} = 10^{8}

**Step 3 :**

Combine the results of step 1 and step 2 to write the final answer in scientific notation.

85.44 x 10^{8} = 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 10^{8}.

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.

10^{14} ÷ 10^{8} = 10^{14-8}

10^{14} ÷ 10^{8} = 10^{6}

**Step 4 :**

Combine the answers to write the rate in scientific notation.

0.9 x 10^{6} = 9.0 x 10^{5}

**Justify and Evaluate : **

Use estimation to check the reasonableness of your answer.

9.0 x 10^{5} is close 10^{6}, so the answer is reasonable.

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