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Chapter 1: Problem 13

Find the value of each exponential expression. $$ 4^{3} $$

Short Answer

Expert verified
The value of \(4^{3}\) is 64.

Step by step solution

01

Understanding the Base and Exponent

Identify the base and exponent in the expression. For the expression \(4^{3}\), the base is 4 and the exponent is 3.
02

Interpreting the Exponent

The exponent indicates how many times the base is multiplied by itself. In this case, 4 is multiplied by itself 3 times.
03

Performing the Multiplication

Multiply the base number by itself as many times as indicated by the exponent. Calculate \[4 \times 4 \times 4 = 64\].
04

Final Answer

The value of the exponential expression \(4^{3}\) is 64.

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Key Concepts

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

Base and Exponent
In exponential expressions, understanding the roles of the base and exponent is crucial. The base is the number that gets multiplied, while the exponent tells you how many times to multiply the base by itself. For instance, in the expression \(4^{3}\), 4 is the base, and 3 is the exponent. The base serves as the foundational value, and the exponent provides direction on the multiplication process. Grasping these two elements is fundamental before proceeding with any calculation involving exponentiation.
Exponentiation
Exponentiation is the mathematical process used to raise a number (the base) to a power (the exponent). This operation is more than just repetitive multiplication; it allows numbers to grow exponentially, which means very quickly! For example, with \(4^{3}\), exponentiation tells us to multiply 4 by itself three times. This results in \[4 \times 4 \times 4 = 64\]. Exponentiation is crucial in many areas of math and science because it models situations where growth or decay happens at an accelerating rate.
Multiplication Process
The multiplication process in exponential expressions is straightforward once the base and exponent are identified. Let鈥檚 dissect \(4^{3}\). Here, start with the base number 4. The exponent 3 tells us to multiply 4 by itself three times. Break it down step-by-step:
  • 4 multiplied by 4 equals 16
  • then, take 16 and multiply it by 4 again
The result is 64. This multiplication process is key in finding the value of the exponential expression efficiently. Understanding each step ensures there are no mistakes along the way.
Exponential Notation
Exponential notation is a compact way to represent repeated multiplication of the same number. Instead of writing out long multiplication chains, you use the base and exponent. For example, instead of writing 4 multiplied by itself three times as \[4 \times 4 \times 4\], you write it concisely as \(4^{3}\). This notation saves space and time and makes it easier to read and write complex expressions. It鈥檚 especially useful when dealing with very large or very small numbers, as it simplifies both understanding and computation.

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