Problem 51 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. The sun is about \(1 \times 10^{8}\) meters from earth.

Step-by-Step Solution

Answer

Question: Convert the scientific notation \(1 \times 10^{8}\) meters, which represents the approximate distance from the Earth to the Sun, into standard decimal notation. Answer: The distance from the Earth to the Sun in standard decimal form is 100,000,000 meters.

Step 1: Identify the coefficient, base, and exponent

The given number in scientific notation is \(1 \times 10^{8}\). The coefficient is 1, the base is 10, and the exponent is 8.

Step 2: Multiply the coefficient by the base raised to the power of the exponent

To convert the number into standard decimal form, we need to multiply the coefficient (1) by the base (10) raised to the power of the exponent (8). This can be written mathematically as \(1 \times 10^8\).

Step 3: Calculate the result

Calculate the result by solving for the base raised to the exponent: \(10^8\). \(10^8 = 100,000,000\). Now multiply the coefficient (1) by the result: \(1 \times 100,000,000 = 100,000,000\)

Step 4: Write the answer in standard decimal form

The distance from the Earth to the Sun in standard decimal form is 100,000,000 meters.

Key Concepts

Decimal Conversion

Decimal conversion is the process of transforming a number into its decimal form. In our case, it involves taking a number from scientific notation and converting it to a standard decimal number.

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. However, sometimes you need to see the number in its regular decimal form, which might be easier to understand or use in some situations.

To convert from scientific notation to decimal form, follow these steps:

• Identify the coefficient, which is the integer part of the scientific notation.
• Determine the exponent of the base 10, which tells us how many places to move the decimal point.
• Move the decimal point according to the exponent: if positive, move to the right; if negative, move to the left.

The number from the exercise, \(1 \times 10^8\), has a coefficient of 1 and an exponent of 8. This means you move the decimal point 8 places to the right, resulting in \(100,000,000\). Versatile knowledge of decimal conversion supports many math problems and real-life applications.

Exponents

Exponents are a way to express repeated multiplication of the same number. They are vital in scientific notation because they simplify the expression of very large or small numbers by showing how many times the base number is multiplied by itself.

In our example, the base is 10 and the exponent is 8. This is written as \(10^8\), meaning that 10 is multiplied by itself 8 times:
10 × 10 × 10 × 10 × 10 × 10 × 10 × 10.

Exponents have a few key properties that help when calculating or simplifying expressions:

• Multiplication: When multiplying numbers with the same base, add their exponents.
• Division: When dividing numbers with the same base, subtract their exponents.
• Power to a Power: Multiply the exponents when raising an exponent to another power.

Understanding how exponents work is crucial for performing decimal conversions in scientific notation.

Standard Form

Standard form or standard decimal form presents numbers as they are measured or used in daily life without exponential notation. This is especially useful for making sense of a number's magnitude at a glance. The process involves converting numbers from scientific notation to their expanded numeric form.

Scientific notation's purpose is to make very large or small numbers more manageable, but when precision or clarity is needed, standard form is preferred. For example, the scientific notation \(1 \times 10^8\) becomes the standard form \(100,000,000\).

Steps to find standard form from scientific notation are:

• Recognize the coefficient and multiply it by the base raised to the exponent, just as demonstrated with \(1 \times 10^8\).
• The expanded number is the standard form, providing a clear understanding of its true value.

Having the ability to convert and use standard form effectively ensures clear communication of values across various fields such as engineering, finance, and sciences.