Problem 55 - Contemporary Precalculus

Question

Express the given number in normal decimal notation. Average distance from the earth to the sun: \(1.50 \times 10^{11}\) meters

Step-by-Step Solution

Answer

Answer: The average distance from the earth to the sun in normal decimal notation is 150,000,000,000 meters.

Step 1: Understand the scientific notation

In scientific notation, a number is expressed as the product of a coefficient (a number between 1 and 10) and a power of 10. In this case, the coefficient is 1.50 and the power of 10 is \(10^{11}\).

Step 2: Multiply the coefficient by the power of 10

To express the given number in normal decimal notation, we have to multiply the coefficient by the power of 10. In this case, we need to multiply 1.50 by \(10^{11}\), which means moving the decimal point 11 places to the right.

Step 3: Write the number in normal decimal notation

After multiplying the coefficient by the power of 10, we get:$$1.50 \times 10^{11} = 150,000,000,000$$ So, the average distance from the earth to the sun is 150,000,000,000 meters in normal decimal notation.

Key Concepts

Decimal Notation

Decimal notation is a way of writing numbers where each digit has a position that represents a power of 10. It's the standard way of writing numbers that you likely use every day.
For example, the number 345.67 in decimal notation means:

• 3 hundreds (which is 3 × 10²)
• 4 tens (which is 4 × 10¹)
• 5 ones (which is 5 × 10⁰)
• 6 tenths (which is 6 × 10⁻¹)
• 7 hundredths (which is 7 × 10⁻²)

Each digit is a place value, with the decimal point separating the whole number part from the fractional part. Decimal notation is crucial in everyday calculations and is used extensively with money and measurements.
Understanding how to express numbers from scientific notation into decimal notation makes it easier to comprehend their size at a glance.

Powers of Ten

The concept of powers of ten is fundamental in understanding scientific notation. A power of ten represents how many times ten is used as a factor. For instance, 10 raised to the power of 2 (written as 10²) means 10 is used as a factor twice, which equals 100.
You might often encounter large or small numbers which are easier to express using powers of ten:

• 10³ is 1,000
• 10⁻² is 0.01

This concept simplifies writing both very large and very small numbers by converting them into a regular and manageable format.
For our example of the Earth's distance to the sun, the power of ten is 10¹¹, signifying you must move the decimal point 11 places to provide the correct decimal notation. This is a quick and efficient way to handle cumbersome numbers in multiple zeroes.

Multiplying Numbers in Scientific Notation

Multiplying numbers in scientific notation involves mainly two steps: multiplying the coefficients and adding the exponents of powers of ten. Let's break it down.
Consider the expression: \(a \times 10^{m} \times b \times 10^{n}\)

• First, multiply the coefficients \(a\) and \(b\)
• Then, add the exponents \(m\) and \(n\)

For example, multiplying 1.50 by 10¹¹:

• The coefficient is 1.50
• The power of ten is 10¹¹

To convert it into full decimal notation, you move the decimal 11 places to the right, resulting in 150,000,000,000.
This method simplifies complex multiplications by dealing separately with parts, especially useful in physics and engineering for handling extremely large or small measurements.