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Chapter 10: Problem 56

In the following exercises, simplify each expression with exponents. $$10^{3}$$

Short Answer

Expert verified
The simplified expression is 1000.

Step by step solution

01

Identify the Base and Exponent

In the expression, the base is 10 and the exponent is 3.
02

Understand What the Exponent Represents

The exponent 3 indicates that the base 10 should be multiplied by itself 3 times.
03

Perform the Multiplication

Multiply 10 by itself 3 times: \(10 \times 10 \times 10 = 1000\).

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Key Concepts

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

Exponents
An exponent is a mathematical notation that indicates how many times a base is multiplied by itself. In the expression \(10^3\), 3 is the exponent. This tells us to multiply the base, 10, by itself three times. Exponents make it easier to write and work with large numbers and repeated multiplication.
Here are some important points about exponents:
  • An exponent of 1 means the base remains unchanged (e.g., \(10^1 = 10\)).
  • Any base raised to the power of 0 is 1 (e.g., \(10^0 = 1\)).
  • Negative exponents represent division (e.g., \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\)).
Understanding these basic rules helps in simplifying more complex expressions involving exponents.
Multiplication
One of the key operations when working with exponents is multiplication. In the expression \(10^3\), we multiply the base 10 by itself three times.
Let's break it down:
  • First multiplication: 10 脳 10 = 100
  • Second multiplication: 100 脳 10 = 1000
Therefore, using an exponent helps us quickly understand that \(10^3 = 1000\). The multiplication method makes these operations straightforward, especially for larger numbers. It's essential to grasp multiplication fundamentals to effectively work with exponents.
Base and Exponent
When dealing with expressions that include exponents, it's vital to identify the base and the exponent.
In the expression \(10^3\),
  • The base is 10. This is the number being multiplied.
  • The exponent is 3. This indicates how many times the base is multiplied by itself.
Recognizing the base and exponent is crucial because it allows you to simplify expressions correctly. For example, knowing that \(10^3\) means multiplying 10 three times helps you arrive at the correct answer, which is 1000. This concept applies to any expression with exponents, making it a fundamental building block in mathematics.

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