Problem 27 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(2.0 \times 10^{-5}\)

Step-by-Step Solution

Answer

\(2.0 \times 10^{-5}\) is \(0.00002\) in standard notation.

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is expressed as the product of a number and a power of 10. In the given number, \(2.0 \times 10^{-5}\), \(2.0\) is the coefficient and \(10^{-5}\) indicates the power of 10.

Step 2: Interpret the Exponent

The exponent \(-5\) in \(10^{-5}\) suggests you need to move the decimal point 5 places to the left. A negative exponent results in a smaller number, moving the decimal to the left increases the number of decimal places to the left of the number.

Step 3: Move the Decimal

Start with the coefficient \(2.0\). Moving the decimal 5 places to the left involves adding zeros to the left of 2: \(0.00002\). Thus, the number is transformed from \(2.0\) to \(0.00002\) by shifting the decimal point 5 places to the left.

Step 4: Write in Standard Notation

After moving the decimal point, the number in standard notation is \(0.00002\). No digits follow the 2, except zeros preceding it before reaching the decimal point. Thus, the scientific notation \(2.0 \times 10^{-5}\) is \(0.00002\) in standard form.

Key Concepts

Standard Notation

Standard notation is a way of writing numbers using our regular counting numbers, without any powers of ten. This is the way we commonly write numbers in daily life. For instance, when you see the number 1000 instead of its scientific form, this is standard notation.

When converting from scientific notation to standard notation, like in the exercise where you convert \(2.0 \times 10^{-5}\) to \(0.00002\), it’s important to use the decimal system we are all familiar with. Here, the goal is to express the number straight, without exponential parts, which makes it easier to read in everyday contexts.

Exponent Interpretation

An exponent in scientific notation indicates how many times you need to move the decimal point. This is a powerful tool for dealing with very large or very small numbers.

For example, a positive exponent, like \(10^5\), means you move the decimal point 5 places to the right. This action increases the value of the number by adding zeros to its end. Conversely, a negative exponent, such as \(10^{-5}\) in our example \(2.0 \times 10^{-5}\), means moving the decimal point 5 places to the left.

• A positive exponent results in a larger number.
• A negative exponent results in a smaller number.

Understanding the role of the exponent is essential for mastering how scientific notation can be converted into standard notation.

Decimal Movement

Decimal movement is the process of repositioning the decimal point in a number, guided by the exponent in scientific notation. If we take the number \(2.0 \times 10^{-5}\), the exponent \(-5\) indicates moving the decimal to the left five places.

This involves starting with the number 2.0 and continually shifting the decimal to the left, for every digit indicated by the exponent. Here’s how it is done:

• Begin with 2.0.
• Move the decimal once, making it 0.20.
• Move it again for 0.020, and continue until you reach 0.00002.

Every move away from the original position involves inserting a zero. This process transforms the scientific notation into an easily recognizable standard form.