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Chapter 4: Problem 79

Explain how to evaluate \(7^{3}\).

Short Answer

Expert verified
The evaluated result of \(7^{3}\) is 343.

Step by step solution

01

Identify Base and Exponent

Identify the base which is 7, and the exponent which is 3.
02

Write Down the Multiplier

Understand that the base number 7 is to be multiplied by itself 3 times. Hence, write the multiplication expression as \(7 \times 7 \times 7\).
03

Multiply

Now, perform the multiplication operation. First, multiply the first two 7's to get 49 and then multiply 49 by the last 7 to get 343.

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Key Concepts

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

Base and Exponent
In mathematics, when you see an expression like \(7^3\), it is essential to understand the components involved. The expression is known as a power, consisting of two parts:
  • Base: The base is the number that is being multiplied. In our example, 7 is the base.
  • Exponent: The exponent tells us how many times the base is used as a factor. In this case, 3 is the exponent.
The power simply tells us that we need to multiply the base by itself as many times as the number represented by the exponent.
In our example of \(7^3\):
  • The base "7" needs to be used three times in the multiplication process.
Understanding these two components makes it easier to work with exponentiation in various mathematical contexts.
Multiplication
Multiplication is a fundamental mathematical operation that involves combining equal groups. From our power example, multiplying involves performing repeated additions of the same number. For \(7^3\), this can be broken down further:
  • First, we multiply 7 by 7 which involves adding seven 7s (\(7 + 7 + 7 + 7 + 7 + 7 + 7\)), equaling 49.
Next, the product (49) is then multiplied by the base one last time:
  • Multiply 49 by 7.
  • This results in 343.
Thus, through multiplication, we've effectively calculated \(7^3 = 343\). This method of multiplying powers is at the heart of understanding exponents.
Mathematical Operations
Mathematical operations are procedures or steps followed to solve expressions or equations. In our example solving \(7^3\), we used one of these key operations: multiplication.
Generally, when dealing with exponents, you are performing a sequence of multiplications. This shows how operations, specifically repeated multiplication, are related through exponents. It involves:
  • Identifying the operation(s) involved.
  • Performing the operation according to the rules of order (e.g., Parentheses, Exponents, Multiplication, Division, Addition, Subtraction - PEMDAS).
Understanding mathematical operations and their properties is crucial, especially when handling more complex expressions and equations in algebra and beyond. The clarity in following these operations helps prevent mistakes and allows for a correct evaluation of expressions.

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