Problem 58 - Prealgebra

Question

Write each number in scientific notation. $$52,000,000$$

Step-by-Step Solution

Answer

The scientific notation is \(5.2 \times 10^7\).

Step 1: Identify the Significant Figures

The first step in converting a number to scientific notation is to identify the significant figures in the number. For 52,000,000, the significant figures are 5.2.

Step 2: Determine the Power of 10 Required

Next, we need to determine how many places to move the decimal point so that the number is a single digit between 1 and 9 to the left of the decimal point. Moving the decimal 7 places to the left, we convert 52,000,000 into 5.2.

Step 3: Form the Scientific Notation

Now that we have 5.2 from the previous steps, we express the number in scientific notation as the significant figures multiplied by 10 raised to the power of the count of decimal movements. Thus, 52,000,000 is written as \(5.2 \times 10^7\).

Key Concepts

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision and accuracy. When writing a number in scientific notation, it is crucial to identify which digits are meaningful. In the case of 52,000,000, the digits '52' are considered significant figures because they represent the precision of the number. The zeroes that follow are placeholders and do not add to the precision, so they aren't significant.

Significant figures help communicate the accuracy of a number, especially when dealing with very large or very small values. Remember these key points about significant figures:

• All non-zero digits are always significant.
• Any zeros between significant digits are also significant.
• Leading zeros before the first non-zero digit are not significant.
• Trailing zeros in a decimal number are significant if there is a decimal point.

Keeping these rules in mind ensures that scientific notation accurately reflects the true precision of a number.

The Concept of Power of 10

The power of 10 represents how many times you multiply a number by 10. When writing in scientific notation, the power of ten shows how many decimal places a number was moved. For example, when converting 52,000,000 into scientific notation, we get 5.2. The decimal originally located after the last zero is moved 7 places to the left, transforming the number into 5.2.

That movement of 7 places to the left is represented by the power of 10 as follows:

• When you move the decimal to the left, the power of 10 is positive.
• If you were moving the decimal point to the right, the power of 10 would be negative.

Hence, moving the decimal in 52,000,000 to get it into scientific form results in the power of 10 being 7, making the expression: \[5.2 \times 10^7\]. This clearly shows the large scale of the number using a compact format.

Navigating Decimal Point Movement

Decimal point movement is a vital concept when expressing numbers in scientific notation, as this action changes the number into a more manageable form. When you move the decimal point, you’re essentially changing the magnitude of the number by factors of 10. To determine the power of 10 in scientific notation, you must count how many places you move the decimal point.

For a number such as 52,000,000, we identify the significant figures 5.2 and need to calculate how to place the decimal to satisfy scientific notation rules. Moving it seven places to the left, the decimal becomes 5.2. This fashioning results in the expression \[5.2 \times 10^7\], highlighting its somber magnitude compared to practical everyday numbers.

• Moving the decimal left increases the power of 10 and signifies a larger positive exponent.
• Moving it right, on the other hand, yields a negative exponent.

Understanding this simple movement helps easily convert regular numbers into an efficient scientific form.