Multiplying and dividing with scientific notation :
We can multiply and divide numbers with scientific notation by using properties of exponents.
Example 1 :
(3.2 x 10⁵) x (26.7 x 10³)
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 2 :
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.
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.
1) Light travels at a speed of 1.86 × 10⁵ miles per second. It takes light from the Sun about 4.8 × 10³ seconds to reach Saturn. Find the approximate distance from the Sun to Saturn. Write your answer in scientific notation.
2) Light travels at the speed of 1.17 × 10⁷ miles per minute. Pluto’s average distance from the Sun is 3,670,000,000 miles. On average, how long does it take sunlight to reach Pluto? Write your answer in scientific notation.
Answers for the above questions :
1) 8.928 × 10⁸ miles
2) 3.14 × 10² minutes
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