Problem 51 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation without using a calculator. Write the result in scientific notation. $$ \frac{3.5 \times 10^{-4}}{5 \times 10^{-1}} $$

Step-by-Step Solution

Answer

The result of the division is \(0.7 \times 10^{-3}\).

Step 1: Divide the coefficients

The coefficients are 3.5 and 5. Divide 3.5 by 5. The result is 0.7.

Step 2: Subtract the exponents

The subtract operation is performed on the exponents. This is -4 minus (-1), which is equivalent to -4 + 1. The result is -3.

Step 3: Combine results

Combine the results from Steps 1 and 2 to express in scientific notation. The final answer is \(0.7 \times 10^{-3}\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics used to represent how many times a number, called the base, is multiplied by itself. Exponents make it easier to write and manage large or small numbers by using exponential notation rather than writing out the entire multiplication. For example, instead of writing 1000, you can write \(10^3\), meaning that 10 is multiplied by itself three times (\(10 \times 10 \times 10\)).
When working with exponents, it's crucial to understand the rules that govern operations such as multiplication and division. For instance, when dividing numbers in scientific notation, you subtract the exponent of the denominator from the exponent of the numerator. This principle helps streamline calculations and keeps numbers in a manageable format.

• Exponents are written as small numbers above the base number.
• They indicate repeated multiplication.
• The subtraction of exponents is key when dividing numbers in scientific notation.

The Role of Coefficients in Scientific Notation

Coefficients are the non-exponential parts of numbers written in scientific notation. In the expression \(3.5 \times 10^{-4}\), the coefficient is 3.5. Think of the coefficient as the first number you see in scientific notation, which can be any number between 1 and 10. It scales the power of ten that follows it, providing the significant figures of the number’s true value.
When performing operations such as multiplication or division with numbers in scientific notation, handle the coefficients separately from the powers of ten. This maintains the accuracy of the number's scale without affecting its exponential part. In division, like in our exercise, you directly divide the coefficients:

• Identify the coefficients from each part of the expression.
• Perform the arithmetic operation (like division).
• Keep the result within the range of 1 to 10 for scientific notation.

Division of Exponents in Scientific Notation

Dividing numbers that are in scientific notation involves handling both the coefficients and exponents separately. After dividing the coefficients, the subtraction of the exponents follows naturally and is central to simplifying the division process.
When you encounter a dividing expression like \(\frac{3.5 \times 10^{-4}}{5 \times 10^{-1}}\), you would:

• First, divide the coefficients: \(\frac{3.5}{5} = 0.7\).
• Then, subtract the exponents of 10: \(-4 - (-1) = -4 + 1 = -3\).

After subtracting the exponents, combine the quotient of the coefficients with the result. Therefore, the final answer, \(0.7 \times 10^{-3}\), effectively demonstrates how division of exponents simplifies complex expressions while maintaining a clear, concise scientific notation.