Problem 73 - Chemistry and Chemical Reactivity

Question

Express the following numbers in exponential or scientific notation. (a) 0.054 (b) 5462 (c) 0.000792

Step-by-Step Solution

Answer

(a) \(5.4 \times 10^{-2}\), (b) \(5.462 \times 10^3\), (c) \(7.92 \times 10^{-4}\)."

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of ten. It looks like this: \( a \times 10^n \) where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Convert 0.054 to Scientific Notation

To convert 0.054, shift the decimal point to the right until you have a number between 1 and 10. This gives 5.4. Since we moved the decimal 2 places to the right, the exponent will be negative: \( 5.4 \times 10^{-2} \).

Step 3: Convert 5462 to Scientific Notation

For 5462, move the decimal point 3 places to the left to get a number between 1 and 10, which yields 5.462. The exponent will be positive: \( 5.462 \times 10^3 \).

Step 4: Convert 0.000792 to Scientific Notation

For 0.000792, shift the decimal point 4 places to the right to get 7.92. The exponent is negative: \( 7.92 \times 10^{-4} \).

Key Concepts

Exponents

Exponents in mathematics are a way to express repeated multiplication of a number by itself. For example, if you see an expression like \(10^3\), it means 10 multiplied by itself three times: \(10 \times 10 \times 10 = 1000\).
Exponents are useful when dealing with very large or very small numbers, as they allow writing these numbers in a more compact form. In scientific notation, exponents are used to indicate the power of ten that the base number is multiplied by.
This notation helps simplify dealing with such numbers, especially in fields like science and engineering, where precision is important. Exponents play a crucial role in understanding how many times the decimal point has been moved in a number placed in scientific notation.

Decimal Point

The decimal point is a dot (.) used to separate the integer part of a number from its fractional part. Moving the decimal point is a common step when converting a regular number into scientific notation.
For example, when converting a number like 0.054, the decimal point is moved two places to the right to get 5.4. This adjustment is necessary to get a number between 1 and 10, which is required for the scientific notation.

• When the decimal point moves to the right, the exponent becomes negative.
• When it moves to the left, the exponent is positive.

Understanding where to place the decimal point and how its movement affects the exponent is essential in mastering scientific notation.

Powers of Ten

Powers of ten are an integral part of scientific notation. They indicate how many times we multiply or divide by ten, based on the exponent's sign and value.
A power of ten is usually written as \(10^n\), where \(n\) can be any integer. The value of \(n\) determines how far the decimal point moves:

• If \(n\) is positive, the decimal point shifts to the right, increasing the number.
• If \(n\) is negative, the decimal point shifts to the left, decreasing the number.

For instance, in \(5.4 \times 10^{-2}\), the \(10^{-2}\) shows that the number is much smaller than 5.4. Conversely, \(5.462 \times 10^3\) indicates that the number is much larger than 5.462.
These powers of ten help convert numbers into a manageable form, making calculations easier and reducing the potential for error.