Problem 52 - Precalculus: Functions and Graphs

Question

Express the number In decimal form. (a) \(2.3 \times 10^{7}\) (b) \(7.01 \times 10^{-9}\) (c) \(1.23 \times 10^{10}\)

Step-by-Step Solution

Answer

(a) 23000000 (b) 0.00000000701 (c) 12300000000

Step 1: Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers in the form of \( a \times 10^{n} \), where \( a \) is a number between 1 and 10, and \( n \) is an integer. To convert the number to decimal form, you simply move the decimal point to the right if \( n \) is positive or to the left if \( n \) is negative, according to the value of \( n \).

Step 1: Convert \(2.3 \times 10^{7}\) to Decimal Form

The exponent \( n \) is 7, which means we move the decimal point 7 places to the right from the position after the 2 to get the number in standard form. So, \( 2.3 \cdot 10^{7} = 23000000. \)

Step 2: Convert \(7.01 \times 10^{-9}\) to Decimal Form

The exponent \( n \) is -9, indicating that the decimal point should be moved 9 places to the left. This results in \( 0.00000000701 \). This number is expressed with 9 zeros after the decimal point before 701.

Step 3: Convert \(1.23 \times 10^{10}\) to Decimal Form

For \( 1.23 \times 10^{10} \), the exponent 10 tells us to move the decimal point 10 places to the right. This gives us \( 12300000000 \).

Key Concepts

Decimal Form Conversion

When numbers are provided in scientific notation, converting them to decimal form helps in visualizing their full scale. Scientific notation often involves the format \( a \times 10^{n} \), where \( a \) is a number between 1 and 10, and \( n \) is an exponent. To get the decimal form:

• If \( n \) is positive, the decimal point in \( a \) moves to the right by \( n \) steps.
• If \( n \) is negative, the decimal point shifts to the left by \( n \) steps.

Breaking it down further:- For example, \( 2.3 \times 10^{7} \) involves moving the decimal 7 places to the right, resulting in 23,000,000.
- On the other hand, \( 7.01 \times 10^{-9} \) involves moving the decimal 9 places to the left, resulting in 0.00000000701.
Thus, converting scientific notation to decimal form makes large or small numbers readily understandable.

Exponentiation

Exponentiation is the mathematical operation used to describe numbers raised to a power. In scientific notation, a number is written as \( a \times 10^{n} \). Here:

• The base is 10, the standard base for decimal numbers.
• The exponent \( n \) signifies how many times 10 is multiplied by itself.

Consider \( 1.23 \times 10^{10} \): The exponent 10 indicates multiplying 1.23 by 10 ten times. This results in 12,300,000,000.
Exponentiation in scientific notation effectively helps represent numbers by compactly expressing the magnitude through powers of ten.
Understanding exponentiation is key in effectively working with large or small numbers in scientific notation.

Large Numbers Representation

Scientific notation is especially useful for representing large numbers compactly and clearly. It simplifies calculations and helps avoid errors in reading and writing zeros. For instance:

• \( 2.3 \times 10^{7} \) transforms into 23,000,000, quickly portraying its magnitude without counting many zeros.
• Similarly, \( 1.23 \times 10^{10} \) becomes 12,300,000,000, compactly encoded as a power of 10.

Using scientific notation, scientists and engineers can process big data and measurements easily and without the tedium of managing countless zeros. Large numbers find familiarity in contexts like astronomy, where distances are vast, or technology, where data sizes can be immense.

Small Numbers Representation

Just as scientific notation excels at expressing large numbers, it is equally adept at representing small numbers. This is crucial in fields where precision is important, such as chemistry or physics. For example:

• The expression \( 7.01 \times 10^{-9} \) neatly represents 0.00000000701 without lengthy zeros before significant digits.

Instead of struggling with trailing zeros, small numbers in scientific notation maintain clarity and precision.
This concise form saves space and ensures that the true scale of tiny quantities is captured, allowing practitioners to communicate effectively about miniscule values.
Whether it's the size of particles or minute concentrations, scientific notation makes small numbers manageable and easier to interpret.