## Lesson 4: Scientific Notation Answer Key

### Introduction

Scientific notation is a mathematical representation used to express numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in scientific and mathematical calculations, as well as in various fields such as physics, chemistry, and engineering. In this lesson, we will explore the concept of scientific notation and provide an answer key to help you practice and understand its application.

### Understanding Scientific Notation

Scientific notation is a way to express numbers in the form of a × 10^b, where "a" is a number between 1 and 10, and "b" is an integer representing the power of 10. This form allows us to easily represent very large or very small numbers by using a concise and standardized format.

### Converting Numbers to Scientific Notation

To convert a number to scientific notation, follow these steps:

- Identify the non-zero digits in the original number.
- Count the number of places you need to move the decimal point to the left or right to obtain a number between 1 and 10.
- Write the number in the form of a × 10^b, where "a" is the number obtained in step 2, and "b" is the number of places you moved the decimal point.

### Example 1: Converting a Large Number to Scientific Notation

Let's consider the number 45,000,000. To convert it to scientific notation:

- The non-zero digits are 4 and 5.
- We need to move the decimal point 7 places to the left to obtain a number between 1 and 10.
- Therefore, the number in scientific notation is 4.5 × 10^7.

### Example 2: Converting a Small Number to Scientific Notation

Now, let's convert the number 0.0000032 to scientific notation:

- The non-zero digits are 3 and 2.
- We need to move the decimal point 6 places to the right to obtain a number between 1 and 10.
- Therefore, the number in scientific notation is 3.2 × 10^-6.

### Performing Operations with Scientific Notation

When performing operations with numbers in scientific notation, it is important to follow certain rules:

**Multiplication:**Multiply the coefficients and add the exponents of 10.**Division:**Divide the coefficients and subtract the exponents of 10.**Addition/Subtraction:**Adjust the exponents of 10 so that they are equal, then add or subtract the coefficients.

### Example 3: Multiplication

Let's multiply two numbers in scientific notation: (3.2 × 10^4) × (1.5 × 10^3).

- Multiply the coefficients: 3.2 × 1.5 = 4.8.
- Add the exponents of 10: 4 + 3 = 7.
- The result is 4.8 × 10^7.

### Example 4: Division

Now, let's divide two numbers in scientific notation: (6.4 × 10^6) ÷ (2 × 10^3).

- Divide the coefficients: 6.4 ÷ 2 = 3.2.
- Subtract the exponents of 10: 6 - 3 = 3.
- The result is 3.2 × 10^3.

### Example 5: Addition

Lastly, let's add two numbers in scientific notation: (2.5 × 10^5) + (1.8 × 10^4).

- Adjust the exponents of 10: 2.5 × 10^5 = 25 × 10^4.
- Add the coefficients: 25 + 1.8 = 26.8.
- The result is 26.8 × 10^4.

### Answer Key

Now, let's go through the answer key for the scientific notation exercises:

- Write the number 0.00000042 in scientific notation:
- Answer: 4.2 × 10^-7.

- Perform the operation (5.6 × 10^3) × (2 × 10^2):
- Answer: 1.12 × 10^6.

- Convert the scientific notation 1.2 × 10^-4 to decimal form:
- Answer: 0.00012.

- Perform the operation (7.8 × 10^5) ÷ (3 × 10^2):
- Answer: 2.6 × 10^3.

- Convert the decimal number 0.0092 to scientific notation:
- Answer: 9.2 × 10^-3.

### Conclusion

Scientific notation is a powerful tool for representing and performing calculations with large and small numbers. By understanding the principles and following the rules of scientific notation, you can simplify complex calculations and express numbers in a concise and standardized format. With the help of this answer key, you can practice and reinforce your understanding of scientific notation. Keep practicing, and soon you'll be a master of scientific notation!