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Chapter 1: Problem 14

Evaluate each exponential expression. $$3^{4}$$

Short Answer

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Step by step solution

01

Understand the Exponential Notation

The expression is given as \(3^{4}\). The number 3 is called the base, and the number 4 is called the exponent. Exponents indicate how many times the base is multiplied by itself.
02

Expand the Exponential Expression

Write out the expression as repeated multiplication. Since the base is 3 and the exponent is 4, we have: \(3 \times 3 \times 3 \times 3\).
03

Perform the Multiplications

First, multiply the first two 3's: \(3 \times 3 = 9\). Then multiply the result by the next 3: \(9 \times 3 = 27\). Finally, multiply 27 by the last 3: \(27 \times 3 = 81\).

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Key Concepts

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

Understanding Exponential Notation
Exponential notation is a way to represent repeated multiplication of the same number. It consists of a base and an exponent. This notation is written as: \(base^{exponent}\). The exponent tells us how many times the base is multiplied by itself. For example, in the expression \(3^{4}\), the base is 3, and the exponent is 4. It means we multiply 3 by itself 4 times. This helps to simplify writing and solving large multiplications.
Base and Exponent
In any exponential expression, there are two main parts: the base and the exponent. The base is the number being multiplied. The exponent tells us how many times to multiply the base by itself. Using our example \(3^{4}\):
  • The base is 3.
  • The exponent is 4.
This means we do:
3 (multiplied by) 3 (multiplied by) 3 (multiplied by) 3.
Understanding what each part represents is crucial to evaluating exponential expressions accurately.
Repeated Multiplication
When dealing with exponents, we're essentially dealing with repeated multiplication. Take \(3^{4}\), for instance. The exponent indicates that we multiply the base number (3) by itself multiple times:
3 脳 3 脳 3 脳 3. Let's break this down:
  • First, multiply the first two threes: \(3 \times 3 = 9\).
  • Next, take the result and multiply by the next three: \(9 \times 3 = 27\).
  • Finally, multiply 27 by the last three: \(27 \times 3 = 81\).
By following this step-by-step multiplication, we find that \(3^{4} = 81\). Repeated multiplication helps us understand how exponents simplify long multiplication processes.
Step-by-Step Solution
Let's break down the process of evaluating \(3^{4}\) in an easy-to-follow, step-by-step manner:
  • Step 1: Write down the expression: \(3^{4}\). The base is 3, and the exponent is 4.
  • Step 2: Expand the expression to show repeated multiplication: 3 脳 3 脳 3 脳 3.
  • Step 3: Perform the multiplications in sequence:
    • First multiplication: \(3 \times 3 = 9\).
    • Second multiplication: \(9 \times 3 = 27\).
    • Third multiplication: \(27 \times 3 = 81\).
So, the value of \(3^{4}\) is 81. Following this step-by-step process ensures you can systematically solve exponential expressions.

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