Question

When a large or small number is written in standard scientific notation, the number is expressed as the product of a number between 1 and $10,$ multiplied by the appropriate power of $10.$ For each of the following numbers, indicate what power of 10 would be appropriate when expressing the numbers in standard scientific notation. a. 84,961,306 b. 0.4870 c. 0.000033812 d. 3,890,406,000,000

Short answer

The appropriate powers of 10 for each number when expressed in standard scientific notation are: a. 84,961,306: $8.4961306\times {10}^7$ b. 0.4870: $4.870\times {10}^{-1}$ c. 0.000033812: $3.3812\times {10}^{-5}$ d. 3,890,406,000,000: $3.890406\times {10}^{12}$

Step-by-step solution

a. 84,961,306

To write this number in standard scientific notation, we need to rewrite it as a number between 1 and 10, multiplied by the appropriate power of 10. To do this, we can move the decimal point to the left until we are left with a number between 1 and 10. We initially have: $$84,961,306=8.4961306\times .....$$ Now we count how many times we moved the decimal point to the left: - From 84,961,306 to 8,496,130.6 (1 move) - From 8,496,130.6 to 849,613.06 (1 move) - From 849,613.06 to 84,961.306 (1 move) - From 84,961.306 to 8,496.1306 (1 move) - From 8,496.1306 to 849.61306 (1 move) - From 849.61306 to 84.961306 (1 move) - From 84.961306 to 8.4961306 (1 move) So, we moved the decimal point 7 times to the left. Therefore, the appropriate power of 10 is 7, and our final expression is: $$84,961,306=8.4961306\times {10}^7$$

b. 0.4870

For this number, we need to move the decimal point to the right until we have a number between 1 and 10. We initially have: $$0.4870=4.870\times ......$$ Now we count how many times we moved the decimal point to the right: - From 0.4870 to 4.870 (1 move) So, we moved the decimal point 1 time to the right. Therefore, the appropriate power of 10 is -1, and our final expression is: $$0.4870=4.870\times {10}^{-1}$$

c. 0.000033812

For this number, we need to move the decimal point to the right until we have a number between 1 and 10. We initially have: $$0.000033812=3.3812\times ......$$ Now we count how many times we moved the decimal point to the right: - From 0.000033812 to 0.00033812 (1 move) - From 0.00033812 to 0.0033812 (1 move) - From 0.0033812 to 0.033812 (1 move) - From 0.033812 to 0.33812 (1 move) - From 0.33812 to 3.3812 (1 move) So, we moved the decimal point 5 times to the right. Therefore, the appropriate power of 10 is -5, and our final expression is: $$0.000033812=3.3812\times {10}^{-5}$$

d. 3,890,406,000,000

To write this number in standard scientific notation, we need to rewrite it as a number between 1 and 10, multiplied by the appropriate power of 10. To do this, we can move the decimal point to the left until we are left with a number between 1 and 10. We initially have: $$3,890,406,000,000=3.890406\times .....$$ Now we count how many times we moved the decimal point to the left: - From 3,890,406,000,000 to 389,040,600,000 (1 move) - From 389,040,600,000 to 38,904,060,000 (1 move) - From 38,904,060,000 to 3,890,406,000 (1 move) - From 3,890,406,000 to 389,040,600 (1 move) - From 389,040,600 to 38,904,060 (1 move) - From 38,904,060 to 3,890,406 (1 move) So, we moved the decimal point 6 times to the left. Therefore, the appropriate power of 10 is 12, and our final expression is: $$3,890,406,000,000=3.890406\times {10}^{12}$$

Key concepts

Powers of 10

Understanding the concept of "Powers of 10" is crucial when dealing with scientific notation. In mathematics, a power of 10 refers to the number 10 raised to any integer exponent. In essence, it represents how many times the number 10 is multiplied by itself. For example:

• ${10}^3$ is equal to 10 × 10 × 10 = 1000.
• ${10}^{-2}$ is equal to 1 divided by ${10}^2$, which is 0.01.

Using powers of 10 makes it easier to express large and small numbers. It helps to simplify numbers by using these exponents, which can greatly reduce the complexity of calculations. Scientific notation utilizes this property by combining powers of 10 with a decimal number between 1 and 10. This allows for a more manageable representation and calculation of extremely large or tiny values.

Decimal Point Movement

The movement of the decimal point is an essential step in converting a number into scientific notation. The goal is to shift the decimal point to create a number between 1 and 10. Depending on whether you're dealing with a large or small number, you move the decimal point either to the left or to the right.

• For large numbers, you move the decimal point to the left. Each move increases the power of 10 by 1. For example, shifting the decimal in 84,961,306 seven places to the left gives $8.4961306\times {10}^7$.


• For small numbers, such as 0.4870, you move the decimal point to the right. Each move decreases the power of 10 by 1. Thus, 0.4870 becomes $4.870\times {10}^{-1}$ after one move.

Moving the decimal point correctly ensures that when multiplied by the appropriate power of 10, the original value of the number is preserved.

Standard Form

Standard form in scientific notation involves expressing a number as a product of a decimal and a power of 10. This way, the number is always written as $a\times {10}^n$, where $1\le a<10$ and $n$ is an integer which denotes how many places the decimal point has been moved.

• For example, the number 3,890,406,000,000 in standard form is $3.890406\times {10}^{12}$. Here, "3.890406" is the decimal part and "12" indicates the number of places the decimal was shifted to create that decimal.
• Similarly, 0.000033812 converts to $3.3812\times {10}^{-5}$ in standard form.

This method simplifies calculation and representation, especially for those difficult-to-handle very large or very small numbers. By keeping the decimal part between 1 and 10, scientific notation provides consistency and clarity in mathematical communication.