Problem 73 - Algebra and Trigonometry

Question

\(73-80\) . Write each number in scientific notation. $$ 69,300,000 $$

Step-by-Step Solution

Answer

69,300,000 in scientific notation is \( 6.93 \times 10^7 \).

Step 1: Understand Scientific Notation

Scientific notation expresses numbers as a product of a coefficient and a power of 10. In general, the format is \( a \times 10^n \), where \(1 \leq |a| < 10\) is the coefficient and \( n \) is an integer.

Step 2: Identify the Coefficient

For the number 69,300,000, we need to find a coefficient \( a \) such that \( 1 \leq |a| < 10 \). Move the decimal point in 69,300,000 seven places to the left to get \( 6.93 \). So, \( a = 6.93 \).

Step 3: Determine the Power of 10

Since we moved the decimal point 7 places to the left, the power of 10 is 7. Thus, our power of 10 is \( 10^7 \).

Step 4: Write the Number in Scientific Notation

Combine the coefficient and the power of 10 we found: 69,300,000 is written as \( 6.93 \times 10^7 \) in scientific notation.

Key Concepts

Coefficient

In scientific notation, the coefficient is the number that appears before the multiplication sign in the format \(a \times 10^n\). It plays a crucial role in expressing the number because it provides the significant digits that represent the size or magnitude of the number.
For example, if we have the number 69,300,000, to find the coefficient, we need to adjust the number into the \(1 \leq |a| < 10\) range. We do this by moving the decimal point.

• Move the decimal point to the left until only one non-zero digit is left in front of it.
• In this case, moving 7 places gives us 6.93. Therefore, the coefficient is 6.93.

Understanding how to find the coefficient helps us accurately express very large or small numbers in a compact and universally understood way.

Power of 10

The power of 10 in scientific notation helps express how many times the coefficient should be multiplied by ten to get back to the original number. It is represented as \(10^n\), where \(n\) is an integer that indicates how many places the decimal point moved.
For the number 69,300,000, we shifted the decimal point 7 places to the left to get our coefficient (6.93). Thus, the power of 10 here is 7, written as \(10^7\).

• Positive powers indicate numbers larger than one.
• Negative powers signify numbers less than one.

By attaching the correct power of 10, we maintain the value of the original number even though it is now written in a condensed form.

Decimal Point

The decimal point is crucial when converting a number into scientific notation. It marks where the "non-decimal" portion of the number ends and is vital for determining the coefficient and power of 10.
To convert 69,300,000, we would note where the decimal would be (at the end of the number by default). To find the coefficient, we shift this decimal 7 places to the left, resulting in 6.93.

• Moving the decimal point to the left results in a positive power of 10.
• Moving it to the right would yield a negative power of 10.

The process of shifting and counting places helps ensure that the scientific notation correctly represents the original number's magnitude.