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Chapter 2: Problem 43

Evaluate the expression using the product rule, where applicable. $$3^{2} \cdot 3^{2}$$

Short Answer

Expert verified
The expression \(3^{2} \cdot 3^{2}\) can be evaluated using the product rule of exponents, \(a^{m}\cdot a^{n}=a^{m+n}\). Applying the rule, we get \(3^{2+2} = 3^{4}\), which equals 81.

Step by step solution

01

Identify the base and exponents

The expression is of the form \(3^{2} \cdot 3^{2}\), where the base is 3, and both exponents are 2.
02

Apply the product rule

Using the product rule \(a^{m}\cdot a^{n}=a^{m+n}\), our expression becomes \(3^{2+2}\).
03

Add the exponents

Now, we need to add the exponents: \(2+2=4\). So the expression becomes \(3^{4}\).
04

Calculate the final result

Finally, we need to calculate \(3^{4}\), which is equal to \(3\times 3\times 3\times 3=81\). Therefore, the evaluated expression is 81.

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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 fundamental component in mathematics, especially in algebra. They provide a way to express repeated multiplication of the same number. For example, in the expression \(3^2\), the exponent is 2, signifying that the base, which is 3, is to be multiplied by itself once. So, \(3^2 = 3 \times 3\). Exponents simplify complex multiplications, allowing for more concise expression evaluation.
When dealing with exponents, it's crucial to identify both the base and the exponent. This identification helps in applying rules such as the product rule correctly. In general, \(a^n\) means 'a' is multiplied by itself 'n' times, where 'a' is the base and 'n' is the exponent.
The Role of the Base in Expressions
In any expression involving an exponent, the base is the main number being multiplied. In the expression \(3^2\), the base is 3. It's important because this is the number that will be repeatedly used in multiplication as specified by the exponent.
The base doesn't change when you're using the product rule, which allows you to combine powers with the same base. For instance, when you have expressions like \(3^2\) and \(3^2\), since the base is the same (3), you can add the exponents (using the product rule) to simplify your calculations. This concept is especially useful when solving algebraic expressions involving powers.
Expression Evaluation with the Product Rule
Expression evaluation involves simplifying or solving mathematical expressions to find their value. The product rule is a valuable tool in this process when dealing with exponents. It states that when you multiply two powers with the same base, you keep the base and add the exponents. The rule can be written as \(a^m \cdot a^n = a^{m+n}\).
For example, in the expression \(3^2 \cdot 3^2\), both have the same base of 3. Using the product rule, you add the exponents (2+2) to get \(3^4\). Simplifying \(3^4\) further gives 81. This method streamlines the process of working through powers and provides a clearer path to the solution.
Components of an Algebraic Expression
An algebraic expression involves numbers, variables, and operations but often without an equal sign, distinguishing it from equations. It can include terms with exponents which are often simplified using rules like the product rule.
In expressions like \(3^2 \cdot 3^2\), each part of the expression (numbers, symbols, multiplication) plays a role. The product rule, as applied to this expression, helps simplify it into a manageable form (\(3^4\)), which can then be evaluated to get the final result, 81. Managing these components effectively is crucial in solving algebra problems efficiently.

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Most popular questions from this chapter

Simplify the expression using the product rule. Leave your answer in exponential form.

Simplify using the quotient rule. $$\frac{15 w^{2}}{w^{10}}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\left(-1.5 \times 10^{-8}\right)\left(4 \times 10^{6}\right)\)

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{8 \times 10^{7}}{4 \times 10^{4}}\)

Simplify the expression using one of the power rules. $$-8(m n)^{5}$$
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