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Chapter 1: Problem 15

Evaluate. $$ 3^{5} $$

Short Answer

Expert verified
The value of \(3^{5}\) is 243.

Step by step solution

01

Understand the Exponentiation

The expression to evaluate is the exponential expression \(3^{5}\). This means you are asked to multiply the base (3) by itself a number of times equal to the exponent (5).
02

Multiply the Base by Itself

Start by multiplying the base, 3, by itself, 5 times: \(3 \times 3 \times 3 \times 3 \times 3\).
03

Evaluate in Steps

You can simplify the expression step by step:\(3 \times 3 = 9\),\(9 \times 3 = 27\),\(27 \times 3 = 81\),\(81 \times 3 = 243\).

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Key Concepts

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

exponential expression
When dealing with an exponential expression like \(3^{5}\), it means you are working with a mathematical way of expressing repeated multiplication. In this case, the base is 3, and the exponent (or power) is 5. The exponent tells us how many times to use the base in the multiplication.

For example, \(3^{5}\) means 3 multiplied by itself 5 times: \(3 \times 3 \times 3 \times 3 \times 3\). Breaking it down:
  • The base (3) is the number you are multiplying.
  • The exponent (5) tells us how many times to multiply the base by itself.

Knowing this helps you understand what you're actually doing when you see an exponential expression in math problems.
multiplying base
The next step is to multiply the base by itself. This means taking the base (3) and multiplying it repeatedly according to the exponent (5). So, let's start this process step by step:

\[3 \times 3 \times 3 \times 3 \times 3\]

It can feel repetitive, but breaking it down makes it easier:
  • Start with \(3 \times 3\).
  • This gives you 9.
  • Then multiply 9 by another 3, which equals 27.
  • Next, take 27 and multiply it by 3 to get 81.
  • Finally, multiply 81 by 3 to get 243.

This step-by-step multiplication is a crucial part of understanding how exponentiation works.
simplifying multiplication
Simplifying multiplication is the process of breaking down and solving the repeated multiplication in manageable steps. For the expression \(3^{5}\), if you solve it all at once, it can seem overwhelming. Instead, break it into smaller pieces:

1. Solve \(3 \times 3\) first, which equals 9.

2. Then do \(9 \times 3\), resulting in 27.

3. After that, solve \(27 \times 3\), giving you 81.

4. Finally, do \(81 \times 3\), leading to the answer of 243.

Here's a step-by-step breakdown:
  • \(3 \times 3 = 9\)
  • \(9 \times 3 = 27\)
  • \(27 \times 3 = 81\)
  • \(81 \times 3 = 243\)
Simplifying multiplication this way helps you keep track of your steps and understand how the final answer is reached.

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