Problem 65 - Algebra 1

Question

MULTIPLE CHOICE Evaluate \(\frac{1.1 \times 10^{-1}}{5.5 \times 10^{-5}}\) using scientific notation. $$ \begin{array}{lll} \text { (A) } 0.2 \times 10^{-4} & \text { (B) } 2.0 \times 10^{4} \end{array} $$ $$ \begin{array}{llll} \text { (C) } & 2.0 \times 10^{3} & \text { (D) } 0.2 \times 10^{4} \end{array} $$

Step-by-Step Solution

Answer

(C) \( 2.0 \times 10^{3} \)

Step 1: Divide Coefficients

Firstly, divide the coefficients 1.1 and 5.5. This will give us: \( \frac{1.1}{5.5} = 0.2 \)

Step 2: Handle Exponents

The second step is to manage the exponents of 10 by subtracting the exponent in the denominator from the exponent in the numerator. Thus, we have \( 10^{(-1-(-5))} = 10^{4} \). Remember, when dividing with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Step 3: Combine Results

Combine the results we got from steps 1 and 2 to get our final answer in scientific notation. Multiply the result from step 1 with the result from step 2: \( 0.2 \times 10^{4} \). It is important to note that this is not in the correct form for scientific notation, where the coefficient is generally preferred to be between 1 and 10.

Step 4: Convert to Correct Format

We should convert the result in step 3 into proper scientific notation. We can convert \( 0.2 \times 10^{4} \) to \( 2.0 \times 10^{3} \) by moving one decimal place to the right on the coefficient and decreasing the power of 10 by 1.

Key Concepts

Division of Exponents

When working with exponents in scientific notation, division involves a simple rule. For bases that are the same, you subtract the exponent in the denominator from the exponent in the numerator.
For example, in our exercise, we have the expressions \(10^{-1}\) and \(10^{-5}\). Because the base (10) is the same, we can apply the rule: subtract \(-5\) from \(-1\).
Remember: subtracting a negative is the same as addition. Thus, \(-1 - (-5) = -1 + 5 = 4\).
This means after dividing, our exponent becomes \(10^{4}\).
To summarize:

• Identify the common base (here, 10).
• Subtract the exponents (numerator minus denominator).
• Apply the result to the base.

Always ensure your calculations are accurate, as small errors at this stage can lead to incorrect results.

Coefficient Calculation

Coefficients are an essential part of scientific notation. These are the numbers in front of the power of 10.
In the exercise given, you are required to calculate \(\frac{1.1}{5.5}\).
To do this, divide 1.1 by 5.5. This yields 0.2.
Steps for coefficient calculation:

• Divide the numbers directly.
• Simplify the numerator and the denominator (if possible) before division.
• Perform the arithmetic operation to get the new coefficient (here, 0.2).

It's crucial to ensure this step is done correctly, as the coefficient will impact your final result in scientific notation. Take your time with this straightforward arithmetic to avoid mistakes.

Proper Format for Scientific Notation

Scientific notation is a standard way of expressing very large or very small numbers. In this format, the coefficient must always be a number between 1 and 10.
If after your calculations the coefficient is outside of this range, you will need to adjust it.
Let's examine the result of our exercise, which was \(0.2 \times 10^{4}\). This initial result is not in proper scientific notation format because the coefficient, 0.2, is less than 1.
To adjust, convert it like this:

• Move the decimal in the coefficient one place to the right (0.2 becomes 2.0).
• Decrease the exponent by 1 to counteract the movement (\(10^{4}\) becomes \(10^{3}\)).

This ensures the coefficient is between 1 and 10, converting \(0.2 \times 10^{4}\) into \(2.0 \times 10^{3}\), achieving the correct format in scientific notation.