Problem 25 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(6.789 \times 10^{-2}\)

Step-by-Step Solution

Answer

0.06789

Step 1: Identify the Parts of Scientific Notation

The number is given as \(6.789 \times 10^{-2}\). Here, 6.789 is the coefficient, and \(10^{-2}\) indicates the power of ten.

Step 2: Understand What the Exponent Represents

The exponent \(-2\) indicates how many places we need to move the decimal point to the left. A negative exponent moves the decimal to the left.

Step 3: Move the Decimal Accordingly

Since the exponent is \(-2\), we move the decimal in 6.789 two places to the left. This changes 6.789 to 0.06789.

Key Concepts

Understanding the Coefficient in Scientific Notation

In scientific notation, the coefficient plays a crucial role. It is the number that you multiply by a power of ten. This coefficient is typically a decimal number that is greater than or equal to 1 and less than 10. In the expression \(6.789 \times 10^{-2}\), 6.789 is the coefficient. The coefficient reflects the significant digits of the number and gives an indication of its size without the zeros trailing after or before the decimal. This makes it easier to read and write large or small numbers efficiently.

• Must be between 1 and 10
• Reflects the significant digits
• Easier for computation and readability

By understanding and identifying the coefficient, we can simplify complex numerical expressions and communicate them effectively.

The Role of the Exponent

In scientific notation, the exponent indicates the power of ten by which you multiply the coefficient. This significantly influences whether you're dealing with a large quantity or a minute fraction. In our example \(6.789 \times 10^{-2}\), the exponent is \(-2\). This tells us to adjust the decimal point to the left, essentially dividing the coefficient by 100.

• Positive exponent moves the decimal point to the right
• Negative exponent moves the decimal to the left
• Represents powers of ten in multiplication or division

Exponent not only simplifies calculations but offers a clean, systematic way to handle extreme numbers in a compact form.

Decimal Point Movement in Scientific Notation

The movement of the decimal point is guided by the exponent in scientific notation. It dictates how and where the decimal moves, ensuring that large or small numbers retain their integrity in their representation. For the number \(6.789 \times 10^{-2}\), the negative exponent \(-2\) means the decimal point moves two places to the left, converting it to 0.06789.

• Left movement for negative exponents
• Right movement for positive exponents
• Maintains the number's value in a new form

This systematic movement helps in transitioning from simplified scientific notation to standard numerical form, making interpretation and practical usage much more manageable.