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Chapter 1: Problem 20

Find the value of each exponential expression. $$ 3^{5} $$

Short Answer

Expert verified
243

Step by step solution

01

Understand Exponents

Exponential notation, like \(3^{5}\), indicates that the base (3) is being multiplied by itself a certain number of times (5 times in this case).
02

Multiply the Base

Write out the expression as a series of multiplications: \(3^{5} = 3 \times 3 \times 3 \times 3 \times 3\).
03

Calculate Step-by-Step

First, multiply the first two numbers: \(3 \times 3 = 9\). Next, multiply the result by the next 3: \(9 \times 3 = 27\). Continue multiplying by 3: \(27 \times 3 = 81\). Finally, multiply by 3 once more: \(81 \times 3 = 243\).

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Key Concepts

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

Understanding Exponents
Exponents are a shorthand way of expressing repeated multiplication. When you see an expression like \(3^{5}\), it's telling you to multiply the base (3) by itself the number of times indicated by the exponent (5). So, in this case, we have: \[3^{5} = 3 \times 3 \times 3 \times 3 \times 3.\] The exponent makes it easier to write and work with large numbers without writing out the full multiplication each time. It's like a shortcut for repeated multiplication.
Base and Power in Exponential Expressions
In any exponential expression, there are two main components: the base and the power. The base is the number being multiplied, and the power (or exponent) tells you how many times to multiply the base. For example, in the expression \(3^{5}\):
  • The base is 3.
  • The power (or exponent) is 5.

This means you take the base (3) and multiply it by itself based on the exponent. So, \(3^{5}\) is calculated as: \[3 \times 3 \times 3 \times 3 \times 3 = 243.\] Understanding these two components is crucial when dealing with exponents, as they provide the structure for how exponents work.
Calculating Multiplication in Exponential Expressions
Breaking down the multiplication step-by-step helps ensure that you accurately calculate an exponential expression. Let's look at \(3^{5}\) again and calculate it step-by-step:
  • First, multiply the first two 3s: \(3 \times 3 = 9.\)
  • Then multiply the result by the next 3: \(9 \times 3 = 27.\)
  • Continue by multiplying by another 3: \(27 \times 3 = 81.\)
  • Finally, multiply by the last 3: \(81 \times 3 = 243.\)

This process helps you understand how exponents turn into repeated multiplications, producing larger values. So, any time you see an exponent, just remember to multiply the base by itself as many times as the power indicates.

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Most popular questions from this chapter

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