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+26 Lesson 6 Homework Practice Scientific Notation

Homework 1 Scientific Notation and Exponents (Chapter BioLink
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Lesson 6 Homework Practice Scientific Notation

Introduction

Scientific notation is a powerful tool used in the field of mathematics to express very large or very small numbers in a more concise and manageable format. It is commonly used in scientific and engineering calculations, as well as in everyday life to represent quantities such as distances, masses, and populations. In this lesson, we will practice applying scientific notation to various numerical expressions, reinforcing our understanding of this important concept.

What is Scientific Notation?

Scientific notation, also known as exponential notation, is a way of expressing numbers in the form of a coefficient multiplied by a power of 10. The coefficient is typically a number greater than or equal to 1 but less than 10, and the power of 10 represents the number of decimal places the decimal point must be moved to obtain the original number. For example, the number 3,450,000 can be expressed in scientific notation as 3.45 x 10^6, where the power of 10 is 6 because the decimal point must be moved 6 places to the left to obtain the original number.

Converting Numbers to Scientific Notation

Converting a number to scientific notation involves determining the appropriate coefficient and power of 10. To do this, follow these steps:

  1. Identify the decimal point in the original number.
  2. Count the number of decimal places the decimal point must be moved to obtain a coefficient between 1 and 10.
  3. Write the coefficient, followed by the multiplication symbol ("x"), and then the power of 10.

For example, to convert the number 0.000056 to scientific notation, we first identify the decimal point between the two zeros. Since we need to move the decimal point 5 places to the right to obtain a coefficient between 1 and 10, the scientific notation representation is 5.6 x 10^-5.

Multiplying and Dividing Numbers in Scientific Notation

Multiplying and dividing numbers in scientific notation involves multiplying or dividing the coefficients and adding or subtracting the exponents of 10. To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents. For example, to multiply (3 x 10^4) and (4 x 10^2), we would multiply the coefficients (3 x 4 = 12) and add the exponents (4 + 2 = 6), resulting in (12 x 10^6).

Adding and Subtracting Numbers in Scientific Notation

Adding and subtracting numbers in scientific notation involves aligning the exponents of 10 and then adding or subtracting the coefficients. The exponents of 10 must be equal for the addition or subtraction to be performed. For example, to add (2 x 10^3) and (5 x 10^2), we align the exponents of 10 (2 and 2) and then add the coefficients (2 + 5 = 7), resulting in (7 x 10^2).

Practice Problems

Now that we have reviewed the basics of scientific notation, it's time to put our knowledge into practice. Let's solve some practice problems to solidify our understanding.

  1. Convert the number 0.0000075 to scientific notation.
  2. Convert the number 8,900,000,000 to scientific notation.
  3. Multiply (2 x 10^5) and (3 x 10^3).
  4. Divide (7 x 10^6) by (2 x 10^2).
  5. Add (4 x 10^4) and (6 x 10^3).
  6. Subtract (9 x 10^2) from (1 x 10^3).

Let's work through these problems step by step.

Solution to Problem 1

To convert the number 0.0000075 to scientific notation, we need to move the decimal point 5 places to the right to obtain a coefficient between 1 and 10. The scientific notation representation is therefore 7.5 x 10^-6.

Solution to Problem 2

To convert the number 8,900,000,000 to scientific notation, we need to move the decimal point 9 places to the left to obtain a coefficient between 1 and 10. The scientific notation representation is therefore 8.9 x 10^9.

Solution to Problem 3

To multiply (2 x 10^5) and (3 x 10^3), we multiply the coefficients (2 x 3 = 6) and add the exponents (5 + 3 = 8). The product is therefore 6 x 10^8.

Solution to Problem 4

To divide (7 x 10^6) by (2 x 10^2), we divide the coefficients (7 ÷ 2 = 3.5) and subtract the exponents (6 - 2 = 4). The quotient is therefore 3.5 x 10^4.

Solution to Problem 5

To add (4 x 10^4) and (6 x 10^3), we align the exponents of 10 (4 and 3) and then add the coefficients (4 + 6 = 10). The sum is therefore 10 x 10^4, which can be simplified to 1 x 10^5.

Solution to Problem 6

To subtract (9 x 10^2) from (1 x 10^3), we align the exponents of 10 (2 and 2) and then subtract the coefficients (1 - 9 = -8). The difference is therefore -8 x 10^2, which can be simplified to -8 x 10^2.

Conclusion

Scientific notation is a valuable tool for expressing very large or very small numbers in a more compact and manageable format. By converting numbers to scientific notation and performing arithmetic operations, we can simplify complex calculations and better understand the magnitude of quantities in various contexts. With practice and familiarity, mastering scientific notation will become second nature, enabling us to confidently tackle mathematical challenges in the future.