Problem 11 - Introductory Chemistry Atoms First

Question

Convert \(47.3 \times 10^{-2}\) to standard notation.

Step-by-Step Solution

Answer

The given number \(47.3 \times 10^{-2}\) in standard notation is 0.473.

Step 1: Understand the given number

The given number is \(47.3 \times 10^{-2}\). The term \(10^{-2}\) indicates that we need to move the decimal point two places to the left.

Step 2: Move the decimal point according to the exponent

Since the exponent is -2, we have to move the decimal point 2 places to the left. So, starting from the decimal point, we move it twice to the left: \(47.3 \Rightarrow 4.73 \Rightarrow 0.473\)

Step 3: Write the result in standard notation

The result after moving the decimal point two places to the left is 0.473. So, the given number \(47.3 \times 10^{-2}\) in standard notation is: \(47.3 \times 10^{-2} = 0.473\)

Key Concepts

Decimal Point Movement

When dealing with scientific notation, understanding the movement of the decimal point is crucial. Scientific notation is a method of expressing numbers that are too big or too small conveniently. In this exercise, we start with the number written as \(47.3 \times 10^{-2}\). The term \(10^{-2}\) tells us to move the decimal point. Since the exponent is negative, we move the decimal point to the left.
Imagine the decimal dances based on the exponent's direction and value:

• If the exponent is positive, the decimal point moves to the right.
• If the exponent is negative, as here, the decimal point moves to the left.

For \(10^{-2}\), we move the decimal two places left, transforming \(47.3\) into \(0.473\). This step ensures that we translate our number correctly from scientific to standard notation. Making these movements correctly is key to maintaining the value integrity of the original number.

Exponents

Exponents in scientific notation express how many times to multiply or divide a number by 10. They are small numbers, positioned above and to the right of the 10, like a little pilot steering your decimal point.
An important thing to remember is:

• A positive exponent moves the decimal to the right, showing a multiplication effect.
• A negative exponent moves the decimal to the left, showing a division effect.

For example, the exponent \(-2\) in \(10^{-2}\) signifies that you should divide by 100 (or move two decimal places to the left).
This makes handling very small or very large numbers much easier, as it breaks them down into simpler terms.

Standard Notation

Standard notation is the way we typically write numbers, free of exponents or additional signs, just the plain number with its decimal in a convenient spot. In this exercise, we converted \(47.3 \times 10^{-2}\) to standard notation, which is just \(0.473\).
Standard notation is preferred in everyday use because it's direct and simple. There's no need to think about exponents or powers of ten; you see the number exactly as it is. To reach standard notation from scientific notation, simply follow these steps:

• Identify the direction of decimal movement based on the exponent sign.
• Move the decimal point the number of spaces indicated by the exponent (left for negative, right for positive).

Once completed, the number will be in standard notation format, like our result, \(0.473\). This clarity makes it ideal for immediate use and comparison with other numbers.