Problem 14 - General Chemistry: Principles and Modern Applications

Question

Express each number in common decimal form. (a) \(3.21 \times 10^{-2}\) (b) \(5.08 \times 10^{-4}\) (c) \(121.9 \times 10^{-5}\) (d) \(16.2 \times 10^{-2}\)

Step-by-Step Solution

Answer

The common decimal form of the given numbers are - (a) \(0.0321\), (b) \(0.000508\), (c) \(0.001219\) and (d) \(0.162\).

Step 1: Conversion of \(3.21 \times 10^{-2}\)

Starting with \(3.21 \times 10^{-2}\), move the decimal point two places to the left. The number in common decimal form is \(0.0321\).

Step 2: Conversion of \(5.08 \times 10^{-4}\)

Start with \(5.08 \times 10^{-4}\). Move the decimal point four places to the left. To accommodate for the extra place, a zero is placed at the front. This results in \(0.000508\).

Step 3: Conversion of \(121.9 \times 10^{-5}\)

Next, \(121.9 \times 10^{-5}\) requires moving the decimal point five places to the left. To compensate for the extra places, zeros are added at the front. The resulting number is \(0.001219\).

Step 4: Conversion of \(16.2 \times 10^{-2}\)

Finally, \(16.2 \times 10^{-2}\) requires moving the decimal point two places to the left. The resulting number is \(0.162\).

Key Concepts

Decimal Conversion

Decimal conversion involves changing a number from scientific notation to standard decimal form. Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It consists of a number between 1 and 10 multiplied by a power of 10. For example, if we have the number 3.21 multiplied by 10 to the power of -2, we need to move the decimal point to convert it to decimal form.

When converting, the exponent tells us how many places to move the decimal point:

• A positive exponent means moving the point to the right.
• A negative exponent means moving the point to the left.

For instance, in the example given, moving two places to the left from 3.21 because of the -2 exponent results in 0.0321. Performing this step ensures that the number is transformed accurately into its decimal equivalent.

Exponents

Exponents are a mathematical notation indicating the number of times a number (the base) is multiplied by itself. An exponent is written as a superscript to the right of a number, like this: \[ x^n \]where \( x \) is the base and \( n \) is the exponent. It signifies that \( x \) should be multiplied by itself \( n \) times.

For instance, \( 10^3 \) means 10 × 10 × 10, which equals 1000. Exponents are a useful way to express large numbers or repeated multiplication simply. In scientific notation, the exponents are often powers of 10, which helps in shifting the decimal point appropriately. Whether the result is a large or small number, exponents efficiently and compactly convey the necessary information about the number's value.

Negative Exponents

Negative exponents can seem tricky, but once understood, they are quite simple. When an exponent is negative, it means the base is on the opposite side of the fraction line from where it would be if the exponent were positive. In other words, \[ x^{-n} = \frac{1}{x^n} \]This tells us that the base x needs to be divided, rather than multiplied.

In practical terms of scientific notation, a negative exponent indicates moving the decimal point to the left. So, for a number like \( 10^{-4} \), it means \( \frac{1}{10000} \), or moving the decimal four places to the left for proper conversion to decimal form.

• Negative exponents assist in representing very small numbers effectively.
• This is frequently used in contexts where values are less than one.

In the examples given in the exercise, realizing the power of negative exponents helps in placing zeros before the number, providing clarity on the number's size and scale.