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Chapter 0: Problem 14

Evaluate each exponential expression. $$3^{3} \cdot 3^{2}$$

Short Answer

Expert verified
The simplified result of the expression \(3^{3} \cdot 3^{2}\) is 243.

Step by step solution

01

Applying the Product of Powers Property

This property states that when multiplying terms of the same base, the resulting exponent is the sum of the individual exponents. Therefore, \(3^{3} \cdot 3^{2}\) can be rewritten as \(3^{3+2}\).
02

Simplify the Exponent

Simplify the exponent by adding 3 and 2 together, which gives you 5. Therefore, the expression becomes \(3^{5}\).
03

Calculate the Value of the Expression

Now, it's only a matter of calculating the power. \(3^{5}\) means that 3 is being multiplied by itself 5 times. So, \(3^{5} = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243\).

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Key Concepts

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

Product of Powers Property
When you're working with exponential expressions where the bases are the same but have different exponents, the product of powers property comes in very handy. This property allows you to simplify the multiplication of these terms easily.

Here鈥檚 how it works: If you have an expression like \( a^m \cdot a^n \), the product of powers property says that you can simply add the exponents together, resulting in \( a^{m+n} \).

Why is this useful?
  • It simplifies the expression, so you can perform operations more easily.
  • It maintains the conceptual understanding of how exponentiation works when combining like terms.
In our example \( 3^{3} \cdot 3^{2} \), applying this property gives you \( 3^{3+2} = 3^5 \). Once simplified, you only have one base to work with, making additional calculations straightforward.
Exponentiation
Exponentiation is a mathematical operation involving numbers, where a base is multiplied by itself a certain number of times as indicated by the exponent. In an expression like \( a^n \), \( a \) is the base and \( n \) is the exponent.

Here's what happens:
  • When \( n \) is a positive integer, it tells you how many times to use the base in a multiplication.
  • For example, \( 3^5 \) means multiply 3 by itself 5 times: \( 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \).
Why do we use exponentiation?
  • It's a quick way to represent repeated multiplication.
  • It simplifies reading and writing of large numbers and calculations.
By understanding exponentiation, you can easily manage complex calculations by breaking them down into simpler components involving powers of numbers.
Simplifying Expressions
Simplifying expressions is the process of making a mathematical expression as simple as possible. When simplifying expressions involving exponents, utilizing properties like the product of powers is essential.

Why simplify?
  • It helps you understand and solve problems more efficiently.
  • By reducing complex expressions to simpler forms, you avoid errors and confusion.
In our example, we start with \( 3^3 \cdot 3^2 \). By applying properties like the product of powers, we simplify this to \( 3^5 \). Next, you calculate the value:
  • \( 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \).
Following these steps ensures the process is structured and clear, allowing you to handle larger or more complex expressions with ease.

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Most popular questions from this chapter

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

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