Problem 19 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(9.76 \times 10^{-4}\)

Step-by-Step Solution

Answer

The number in standard notation is 0.00976.

Step 1: Identify the Components

The expression we need to convert is in scientific notation: - The base number is 9.76. - The exponent is -4, which indicates that the decimal point needs to be moved 4 places to the left.

Step 2: Move the Decimal Point

To convert from scientific notation to standard notation, move the decimal point 4 places to the left. - Starting with 9.76, the decimal point moves past the 9 (1 place), then moves three more places, filling with zeros: 0.00976.

Step 3: Write the Final Answer

After moving the decimal point, the number in standard notation is 0.00976.

Key Concepts

Understanding Standard Notation

Standard notation is a way of writing numbers without using exponents. It's the most common way people write numbers daily. When a number is expressed in this form, it means that any decimal present is at its usual position, and no 10s power multiplies are visible. This is ideal for communicating values clearly and concisely.
Here's why it's important to understand:

• It's straightforward for any kind of number, whether very large or very small.
• Without scientific notation, you can easily denote numbers in a form that's easiest for the human eye to interpret directly.
• Although scientific notation is helpful in mathematics or science to simplify complex calculations, converting it to standard notation allows for a more tangible understanding of its size or value in real-world terms.

For instance, when converting from scientific notation, like \(9.76 \times 10^{-4}\), into standard notation, you end up with 0.00976, a clearer representation of the actual number's magnitude.

Converting Scientific Notation to Standard Notation

Scientific notation is incredibly useful for handling extremely large or small numbers due to its concise format. Nonetheless, grasping how to convert it into standard notation can help make sense of these numbers.
Let's talk about the conversion steps:

• Identify the base and the exponent in the scientific notation. In the example \(9.76 \times 10^{-4}\), the base is 9.76, and the exponent is -4.
• The exponent reveals how far and in which direction to move the decimal point. A negative exponent means moving the decimal to the left, while a positive one indicates moving it to the right.
• For \(10^{-4}\), move the decimal 4 places left. Starting with 9.76, shift the decimal past each digit, adding zeros to fill in the vacant places.

After moving the decimal point, you translate the notation into the number 0.00976, offering an immediate understanding of the number's size.

Grasping Exponents in Scientific Notation

Exponents in scientific notation are fundamental to understanding how the number is scaled by powers of ten. The exponent dictates how many zeros follow the base number, should it be enlarged or diminished.
Here's how they function:

• A positive exponent moves the decimal point to the right, making the number larger. For example, \(2.5 \times 10^3\) turns into 2500.
• A negative exponent shifts the decimal point left, effectively turning a large base number into a smaller fraction. In our exercise, \(9.76 \times 10^{-4}\) becomes 0.00976.
• Understanding whether to move left or right depends entirely on the exponent's sign—plus or minus determines direction, while its absolute value tells you how far to move.

Thus, exponents play a key role in determining the number's actual size or position in standard notation.