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Chapter 1: Problem 25

Evaluate. \(3^{5}\)

Short Answer

Expert verified
Answer: The value of the expression \(3^5\) is 243.

Step by step solution

01

Write out the expression

To solve \(3^5\), we will multiply the base (3) by itself 4 more times. So, the expression becomes: \(3 * 3 * 3 * 3 * 3\).
02

Perform the multiplications

Now, we will multiply the numbers together: \(3 * 3 = 9\). After the first multiplication, the expression becomes: \(9 * 3 * 3 * 3\). \(9 * 3 = 27\). After the second multiplication, the expression becomes: \(27 * 3 * 3\). \(27 * 3 = 81\). After the third multiplication, the expression becomes: \(81 * 3\). \(81 * 3 = 243\). We have now completed all the multiplications.
03

Write the final result as an one-line exponentiation expression

The result of evaluating \(3^5\) is 243. Therefore, \(3^5 = 243\).

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Key Concepts

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

Understanding the Base in Exponents
In the world of exponents, the base is the number that is multiplied by itself. It's the foundation of the expression in question. Take for example our exercise, where the base is 3, as seen in \(3^5\). This means that the number 3 is what we are working with, and it's what is being repeated in multiplication.
  • The base can be any real number.
  • Think of the base as the starting block 鈥� it鈥檚 where the process begins.
  • In mathematical terms, the base represents the factor that shows up repeatedly.
The choice of the base can change the entire outcome of the expression, as different bases will lead to different results. A good grasp of understanding bases helps in solving and simplifying exponentiation problems smoothly.
The Role of Multiplication in Exponents
Multiplication is a key operation when it comes to exponents. With exponents, you鈥檙e essentially performing repeated multiplication of the base. When handling \(3^5\), you start with the base 3 and multiply it by itself four additional times: \(3 * 3 * 3 * 3 * 3\).
  • This is different from simple multiplication because it shows how many times you multiply the base by itself.
  • Each step of multiplication progresses you closer to the final product.
  • Understanding this groundwork of multiplication aids in properly calculating the power of any base.
This step-by-step multiplication approach transforms the exponent expression into a clear multiplication sequence making it easier to follow and compute. This is especially useful when handling larger exponents.
Exploring the Concept of Power in Math
In mathematics, power is the term used to denote how many times the base is multiplied by itself. In our example, \(3^5\), the power is 5. This tells us we need five 3鈥檚 to be multiplied together: \(3 \times 3 \times 3 \times 3 \times 3 = 243\). The power component can drastically alter the size of the result:
  • Higher powers lead to larger products.
  • The power is crucial because it defines the height of multiplication.
  • This is conceptually similar to how height or depth defines dimensions.
Getting comfortable with the idea of power, and how it effectively leads to the multiplication of a base, is essential in tackling more complex problems in algebra and beyond.

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