Problem 66 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation. Write the result in scientific notation. (Lesson 8.5). $$ \frac{1.4 \times 10^{-1}}{3.5 \times 10^{-4}} $$

Step-by-Step Solution

Answer

The answer is \(4.0 \times 10^{2}\)

Step 1: Division of Numeric Values

Start with the division of the numeric values: \( \frac{1.4}{3.5} = 0.4 \)

Step 2: Compute the Exponents of Base 10

Then, calculate the difference between the exponents of base 10 which means subtract the exponent of the denominator from the exponent of the numerator: \(-1-(-4) = -1+4 =3 \)

Step 3: Write the Final Answer

Combine the results of step 1 and step 2 to arrive at the answer. Therefore, it becomes \(0.4 \times 10^{3}\). However, this is not the standard form of scientific notation. To convert, you move the decimal point in 0.4 to get 4. and increase the exponent by 1 to maintain the value. Hence the result is \(4.0 \times 10^{2}\)

Key Concepts

Division of Exponents

In scientific notation, numbers are often expressed as a product of a number between 1 and 10, and a power of 10. When dividing such numbers, a critical component is understanding how to manage the exponents associated with these powers of 10.
To divide exponents, you simply subtract the exponent in the denominator from the exponent in the numerator. Consider this example:

• Given: \( \frac{10^{-1}}{10^{-4}} \)
• Subtract the exponents: \(-1 - (-4) = -1 + 4 = 3\)

This step is crucial because it tells us by what power of ten we need to multiply or divide our base number to get the correct scientific notation. By mastering the division of exponents, you unlock the ability to simplify complex calculations in physics, engineering, and many areas of science.

Numeric Division

Numeric division in scientific notation involves dividing the base numbers separately from the powers of ten. In our exercise, you have:

• The division \( \frac{1.4}{3.5} \)
• Calculate to obtain: 0.4

This part of the calculation focuses purely on the division of the numerical coefficients—those numbers that aren't part of the exponent phrase. Numeric division is straightforward arithmetic. By simplifying the numeric part first, it becomes easier to see the overall structure of the problem, making subsequent operations on the exponents less prone to errors. Always aim to carry out this step prior to handling the division of exponents for a clearer path to the solution.

Standard Form in Scientific Notation

Once the division of numbers and exponents is completed, the next task is to ensure the answer is in the standard form of scientific notation. This form requires that the number before the multiplication symbol is between 1 and 10. Let's consider the final steps from our example:
The result after division was \(0.4 \times 10^3\). However, this isn't yet in standard form. To adjust:

• Move the decimal in 0.4 one place to the right, giving us 4.0
• Increase the power of ten by one to compensate, making it \(10^2\)

Thus, the final answer in standard form is \(4.0 \times 10^2\). This conversion is crucial because it maintains the numerical value of the expression, aligning with conventions used in scientific communication. Always ensure that after performing operations, the number is converted accurately into this standard format for clarity and consistency.