Problem 40 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{0.00072}{0.0000024}\)

Step-by-Step Solution

Answer

The result is \( 3 \times 10^{2} \).

Step 1: Convert to Scientific Notation

First, convert each number to scientific notation. The decimal 0.00072 can be written as \( 7.2 \times 10^{-4} \), and the decimal 0.0000024 can be written as \( 2.4 \times 10^{-6} \).

Step 2: Set Up the Division

Set up the division using the numbers in scientific notation: \( \frac{7.2 \times 10^{-4}}{2.4 \times 10^{-6}} \).

Step 3: Divide the Coefficients

Divide the coefficients (the numbers out front): \( \frac{7.2}{2.4} = 3 \).

Step 4: Subtract the Exponents

Using the properties of exponents, subtract the exponent in the denominator from the exponent in the numerator: \( -4 - (-6) = -4 + 6 = 2 \).

Step 5: Combine the Results

Combine the result from dividing the coefficients with the result from subtracting the exponents: \( 3 \times 10^{2} \).

Key Concepts

Understanding the Properties of Exponents

Exponents are a powerful tool in mathematics used to express repeated multiplication. They come with a set of rules known as the properties of exponents, which simplify calculations and make expressions easier to manage. One crucial property is the rule of subtraction for division. When you divide two numbers with the same base and different exponents, you subtract the exponents. For example, \( a^m/a^n = a^{m-n} \). This rule is fundamental in scientific notation calculations, as seen in our problem where we divide \( 10^{-4} \) by \( 10^{-6} \) and subtract the exponents to get \( 10^2 \).

Exponents are not only useful in division but also in multiplication. When multiplying exponents with the same base, you can add the exponents: \( a^m\cdot a^n = a^{m+n} \). Understanding these properties helps in handling scientific notation effectively. It allows you to work with very large or very small numbers more efficiently.

• Zero Exponent Rule: Any base raised to the power of zero equals 1, \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent, \( a^{-n} = 1/a^n \).

Recognizing and applying these rules can significantly simplify the manipulation of expressions with exponents.

Mastering the Division of Exponents

Dividing exponents requires understanding the subtraction property of exponents, which was aptly applied in this exercise. In the scientific notation format, dividing involves working with both the numerical coefficients and the powers of ten.

In the exercise problem, after converting the decimals to scientific notation, you're confronted with \( \frac{7.2 \times 10^{-4}}{2.4 \times 10^{-6}} \). The first step is to divide the coefficients, which are the numbers in front, resulting in 3 as \( 7.2 \div 2.4 \). Next, apply the subtraction rule of exponents to the powers of ten, which gives \( 10^{-4 - (-6)} = 10^2 \). This combines the operations of the division.

The division of exponents therefore relies heavily on subtracting exponents when bases are alike. This property ensures that calculations remain simple and offers a consistent method to follow for solving similar problems.

Converting Decimals to Scientific Notation

Scientific notation is a method of writing very large or very small numbers more succinctly. It uses a base (usually 10) raised to an exponent, paired with a coefficient that is typically between 1 and 10. This notation is particularly useful in scientific calculations as it keeps numbers manageable and easy to work with.

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

• Identify the decimal. For example, 0.00072.
• Move the decimal point to the right of the first non-zero digit, counting how many places you move it. In 0.00072, move the decimal 4 places to the right, resulting in 7.2.
• The number of places moved becomes the negative exponent of 10. Therefore, 0.00072 can be written as \( 7.2 \times 10^{-4} \).

Similarly, for smaller values like 0.0000024, the decimal moves 6 places to give us \( 2.4 \times 10^{-6} \). This helps in converting complex calculations into simpler exponential terms, aiding in efficient and error-free computation.