Problem 140 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$534.7=5.347 \times 10^{3}$$

Step-by-Step Solution

Answer

The statement \(534.7 = 5.347 \times 10^{3}\) is false. The correct statement should be \(534.7 = 5.347 \times 10^{2}\)

Step 1: Understanding scientific notation

Scientific notation is a method to write down very large or very small numbers easily. It's in the form of \(a \times 10^{n}\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer.

Step 2: Convert the left side number into scientific notation

When 534.7 is converted into scientific notation, it becomes \(5.347 \times 10^{2}\).

Step 3: Compare the converted number with the given number on the right side

After conversion, the left side becomes \(5.347 \times 10^{2}\) which is not equal to \(5.347 \times 10^{3}\) that was given on the right side. So, the statement is false.

Step 4: Correct the false statement

As the statement is false, to make it true, we correct the right side to match with the left side after conversion. So, the correct statement becomes \(534.7 = 5.347 \times 10^{2}\)

Key Concepts

Converting Decimals to Scientific Notation

Scientific notation is a handy way to deal with very large or extremely small numbers. It simplifies these numbers into a format that is easier to read and work with. This method uses two important components:

• The coefficient: This is a decimal number greater than or equal to 1 and less than 10.
• A power of 10: This shows how many places the decimal needs to move to transform the number back into its original form.

Let's take the number 534.7. To convert it into scientific notation, follow these steps:

• Move the decimal point left until you have a number between 1 and 10.
• Count how many places you moved the decimal. For 534.7, the decimal is shifted two places to the left, becoming 5.347.
• Write the number as 5.347 multiplied by 10 raised to the power of the number of places you moved the decimal: \[5.347 \times 10^{2}\]

This format helps in making comparisons and calculations simpler.

Comparison of Numbers

Once you've converted numbers to scientific notation, comparing them becomes straightforward. This is because the numbers are neatly arranged into two parts: the coefficient and the power of 10.
Here's how the comparison works:

• First, look at the powers of 10. The number with the higher power is larger. For example, \(5.347 \times 10^3\) is bigger than \(5.347 \times 10^2\) simply because \(10^3\) is greater than \(10^2\).
• If the powers of 10 are the same, then compare the coefficients. The larger coefficient means a larger number. So, if you had \(6.1 \times 10^2\) and \(4.7 \times 10^2\), then \(6.1 \times 10^2\) would be the larger number.

This technique makes it easier to quickly determine size and order of numbers, especially when dealing with numbers of vastly different scales.

Correction of Mathematical Statements

In mathematics, it's crucial to ensure equations and statements are accurate. Whenever you come across a statement like \(534.7 = 5.347 \times 10^{3}\),it's vital to verify each side. If you notice any discrepancies like we did in the example above, here's how to correct it:

• Convert both numbers to scientific notation, if they aren't already. For 534.7, it becomes\(5.347 \times 10^2\).
• Compare both sides based on the scientific notation rules. Identify mismatches (e.g., the power of 10).
• Adjust either side of the equation to accurately reflect equal values. In our case, the statement should be properly corrected to: \[534.7 = 5.347 \times 10^{2}\]

By making these corrections, you ensure precision, which is paramount in all mathematical problems and evaluations.