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Chapter 5: Problem 84

Convert to decimal notation. $$ 8 \times 10^{4} $$

Short Answer

Expert verified
80,000

Step by step solution

01

Identify the Exponential Components

Identify the base and the exponent in the given expression. In this case, the base is 8 and the exponent is 4.
02

Understand the Meaning of the Exponent

The exponent 4 means that we will multiply the base 8 by 10 raised to the power of 4. This implies multiplying 8 by 10,000.
03

Perform the Multiplication

Calculate the result of the multiplication. \ \[ 8 \times 10^4 = 8 \times 10,000 = 80,000 \ \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Components
To begin, let's understand what exponential components are. An expression like \(8 \times 10^{4}\) consists of two main parts:
the base and the exponent. The base is the number being multiplied, which in this case is 8.
The exponent, written as a small number above and to the right of the base, represents how many times the base is multiplied by 10.
This method of expressing large numbers can help simplify mathematical operations and make them easier to handle.
Base and Exponent
The base and exponent form the core elements of exponential notation.
Here, 8 is the base and 4 is the exponent. Let's explore each:
  • **Base:** The base is the number that you are multiplying. In our example, 8 is the base number.
  • **Exponent:** The exponent tells you how many times to multiply the base by itself. In our example, the exponent is 4, which means we have \(10^{4}\).
The expression \(10^{4}\) translates to multiplying '10' by itself four times. It's a shorthand for making calculations more manageable.
Multiplication
Now that we have identified the base and exponent, let's perform the multiplication. The expression \(8 \times 10^{4}\) means we are multiplying 8 by \(10\) raised to the power of 4.
\(10^{4}\) equals 10,000 because \(10 \times 10 \times 10 \times 10 = 10,000\).
So, we multiply:
\[ 8 \times 10,000 = 80,000 \]
Multiplication involving exponents can initially seem complicated, but breaking it down step by step makes it much simpler.
- Start by calculating the exponent result.
- Then, multiply it by the base number.
This approach demystifies exponential notation and makes large numbers approachable.

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