Problem 2 - Elementary Algebra

Question

Write the following numbers in scientific notation. $$ 72.33 $$

Step-by-Step Solution

Answer

Question: Convert the number 72.33 to scientific notation. Answer: The number 72.33 in scientific notation is 7.233 x 10^1.

Step 1: Identify the correct number between 1 and 10

To convert 72.33 to a number between 1 and 10, we need to move the decimal point to the left until there is only one non-zero digit before the decimal point. In this case, we move the decimal point one place to the left, yielding 7.233.

Step 2: Determine the power of 10

Since we moved the decimal point one place to the left, we must multiply the number we obtained in step 1 by 10 raised to the power of 1 to get back to the original number. In this case, 10 raised to the power of 1 is 10.

Step 3: Write the number in scientific notation

Combining the results of steps 1 and 2, the scientific notation for the given number (72.33) is: $$ 7.233 \times 10^1 $$

Key Concepts

Decimal Point Movement

To express a number in scientific notation, it is essential to first adjust its decimal point. This adjustment shifts the number into the standard range between 1 and 10. By moving the decimal point, we identify the new position where our number fits this criterion. In the example of 72.33, the objective is to have only one non-zero digit to the left of the decimal point. Thus, the decimal point is moved one position to the left. This movement gives us 7.233, a number happily nestled between 1 and 10. Whenever you move the decimal point:

• Left: To make the number smaller (if it’s large like 72.33), you count the shifts as positive powers of ten.
• Right: To make the number larger (if it’s small like 0.07233), you count the shifts as negative powers of ten.

Observe this process as the foundational step in transitioning the number from its initial form to one suited for scientific notation.

Power of Ten

Once the decimal point has been moved, determining the power of ten is crucial to converting the original and the new number back and forth reliably. This step ensures the scientific notation representation of the number genuinely reflects its original magnitude.When you move the decimal point, like in our example with 72.33, the direction and the number of places moved come into play:

• Each shift to the left increases the power of ten by 1.
• If you had shifted to the right, the power would be negative, signifying division by ten instead.

In 72.33, the decimal was moved one place left, meaning the power of ten is 1. The scientific notation then becomes an expression like: \(a \times 10^b\), where \(a = 7.233\) and \(b = 1\) for this particular instance.Understanding these powers helps in grasping the scale of numbers in scientific terms, allowing easy comprehension and conversion of large or small numbers aligned in a uniform format.

Standard Form Conversion

Scientific notation is a method of rendering numbers compactly and consistently. To convert a real number into this form, after establishing both the adjusted decimal number and the corresponding power of ten, the number can be rewritten neatly.The standard form for scientific notation is represented as \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. This format simplifies complex numbers for ease of use in calculations and comparisons.For the given number, 72.33:

• We determined \(a = 7.233\) from the decimal adjustment step.
• We discovered \(b = 1\) when calculating the power of ten.

Thus, the fully expressed standard form is \(7.233 \times 10^1\). This notation maintains the integrity and scale of the original number while making it easier to handle, especially in scientific and mathematical contexts. Converting to and from this form helps in precise communication of measurement and calculation without the clutter of unnecessary zeros or digits.