Problem 13 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(2.3 \times 10^{2}\)

Step-by-Step Solution

Answer

230

Step 1: Understanding Scientific Notation

Scientific notation is a way to represent numbers, especially very large or very small numbers. It consists of a number between 1 and 10 multiplied by a power of 10. In this exercise, the number is written as \(2.3 \times 10^{2}\).

Step 2: Write Out The Calculation

To convert the number from scientific notation to standard notation, you need to calculate \(2.3 \times 10^{2}\). This means you multiply 2.3 by 10 raised to the power of 2.

Step 3: Calculate the Power of 10

Calculate \(10^2\), which is 10 multiplied by itself. So, \(10^2 = 10 \times 10 = 100\).

Step 4: Multiply by the Factor

Now, multiply 2.3 by 100: \(2.3 \times 100 = 230\). Thus, \(2.3 \times 10^{2} = 230\).

Step 5: Conclusion

The standard notation for \(2.3 \times 10^2\) is 230.

Key Concepts

Scientific Notation

Scientific notation is a compact way of expressing numbers, particularly those that are extremely large or incredibly tiny. It's especially useful in fields like science and engineering, where such numbers are common. In scientific notation, a number is represented by a decimal part called the mantissa (which should be a value between 1 and 10), and this is multiplied by a power of ten. For example, in the number \(2.3 \times 10^{2}\), 2.3 is the mantissa, and \(10^{2}\) is the power of 10. This notation simplifies calculations and allows us to easily understand and compare different scales of numbers by quickly identifying their order of magnitude. Each component of scientific notation plays a crucial role: the decimal part gives precision, while the power of 10 dictates the scale.

Power of 10

The power of 10, noted as \(10^n\), is a fundamental concept when working with scientific notation. It represents how many times 10 is multiplied by itself.One way to understand this is to think of the exponent, \(n\), as the number of times 10 is used as a factor. For example:

• \(10^1 = 10 \)
• \(10^2 = 10 \times 10 = 100 \)
• \(10^3 = 10 \times 10 \times 10 = 1000 \)

The power of 10 can also be negative, which represents division by 10 multiple times, giving smaller decimal numbers:

• \(10^{-1} = 0.1 \)
• \(10^{-2} = 0.01 \)

Understanding powers of 10 helps us handle numbers of very different sizes without difficulty.

Multiplication

Multiplication in scientific notation involves two main steps: calculating the power of 10 and then multiplying the resulting number by the mantissa.Here's how it works:1. **Calculate the power of 10:** For example, \(10^2\) means multiplying 10 by itself twice, resulting in 100.2. **Multiply the mantissa by this number:** With our example of \(2.3 \times 10^2\), we find \(10^2 = 100\). Then multiply 2.3 by 100.Mathematically, this is shown as:\[2.3 \times 100 = 230\]This multiplication converts the number from scientific notation to standard notation, simplifying it into a form we commonly use for everyday numbers.