Problem 23 - Elementary and Intermediate Algebra

Question

Convert number to standard notation. \(2.718 \times 10^{0}\)

Step-by-Step Solution

Answer

2.718

Step 1: Understanding 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 consists of a number (between 1 and 10) multiplied by a power of ten. Here we have the number 2.718 multiplied by 10 raised to the power of 0.

Step 2: Recognizing the Power of Ten

Any number raised to the power of 0 is 1. That is, for any real number, including 10, we have: \[10^{0} = 1\]

Step 3: Calculating the Expression

Now that we know \(10^{0} = 1\), we multiply 2.718 by 1: \[2.718 imes 10^{0} = 2.718 imes 1 = 2.718\]

Step 4: Conclusion

The standard notation of the given expression \(2.718 \times 10^{0}\) is simply the number itself, 2.718.

Key Concepts

Standard Notation

Standard notation is how we typically write numbers in everyday life. It is the representation of a number without exponents or powers. For instance, when you see a number like 2,500, it is already in standard notation.
Scientific notation, in contrast, is typically used to express very large or very small numbers more succinctly. This involves a coefficient and a power of ten, making it easier to handle in calculations and notation.
Let's break it down:

• In standard notation, 2.718 is already in its simplest form as there are no exponents involved.
• It provides a clearer, straightforward understanding without needing to evaluate powers of ten.

This straightforward representation helps simplify mathematical tasks that do not involve extraordinary ranges of numbers.

Power of Ten

The concept of "power of ten" is central in both scientific and standard notations. A power of ten refers to a number formed by raising ten to an exponent. This exponent indicates how many times ten is multiplied by itself.
For example:

• If the exponent is 1, \(10^{1} = 10\).
• If the exponent is 2, \(10^{2} = 100\).
• If the exponent is 0, \(10^{0} = 1\).

In scientific notation, the exponent dictates the "shifting" of the decimal point. A positive exponent means shifting right, making the number larger, while a negative exponent means shifting left, making the number smaller. In this exercise, we have \(10^{0}\), which equals 1, leaving the main number unchanged. It’s key to understanding expressions like \(2.718 \times 10^{0}\).

Multiplication with Powers of Ten

Multiplication with powers of ten is a method to quickly scale numbers by moving the decimal point. When you multiply a number by a power of ten:

• A positive exponent shifts the decimal point to the right.
• A negative exponent shifts it to the left.
• An exponent of zero means multiplying by 1, leaving the number unchanged.

In this case, multiplying 2.718 by \(10^{0}\) simplifies to multiplying by 1. \[2.718 \times 10^{0} = 2.718 \times 1 = 2.718\]
No change occurs to the decimal placement or value, reiterating the straightforward nature of such a calculation. This concept is helpful in both simplifying problems and verifying answers as it consolidates vast numbers into more manageable forms.