Problem 16 - College Algebra

Question

Evaluate each exponential expression. $$ \left(3^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The value of the exponential expression \((3^{3})^{2}\) is 729.

Step 1: Apply exponent rule

In the expression \((3^{3})^{2}\), we have an exponent (3) raised to another exponent (2). This is where the exponentiation rule comes into play: \((a^m)^n = a^{m \cdot n}\), where a is the base and m and n are the exponents. Therefore, we can simplify the given expression by multiplying the exponents: \((3^{3})^{2} = 3^{3 \cdot 2}\).

Step 2: Simplify the exponent

Now we can simplify \(3 \cdot 2\) to get \(3^{6}\).

Step 3: Evaluate the expression

When we calculate the value of \(3^{6}\), we get the result as 729. This is the final value of the given exponential expression.

Key Concepts

Exponentiation Rule

Understanding the exponentiation rule is crucial when dealing with exponential expressions. It tells us how to handle the situation when we have a power raised to another power, as in the example \( (3^3)^2 \). The rule comes down to a simple multiplication of exponents.

When we have \( (a^m)^n \) where \(a\) is the base and \(m\) and \(n\) are the exponents, the exponentiation rule allows us to rewrite the expression as \( a^{m \cdot n} \), effectively multiplying the exponents.

It's a shortcut that prevents us from having to multiply out the base multiple times, which can quickly become cumbersome with larger exponents. Instead, it delivers a straightforward approach towards simplifying the expression and reaching an answer more efficiently.

Exponents Simplification

Simplifying exponents is a fundamental skill when evaluating expressions like \( (3^3)^2 \). After applying the exponentiation rule, we often end up with a single base raised to a new exponent that's the result of a multiplication. In this case, multiplying \(3 \cdot 2\) to obtain \(3^6\) simplifies our work.

It's essential to understand how to simplify exponents because it's not just about calculation ease. It also helps us observe patterns and properties of numbers, such as growth rates and scale.

Multiplying exponents directly as opposed to expanding them saves time and reduces the likelihood of making errors in longer calculations. This simplicity helps us focus on the underlying concepts rather than the arithmetic itself.

Mathematical Evaluation

Mathematical evaluation, in the context of exponential expressions, is the final step where we actually calculate the value of the expression after applying rules and simplifications. Starting with an expression like \( (3^3)^2 \), after applying the exponentiation rule and simplifying the exponents, we arrive at \(3^6\).

At this point, we process the computation of \(3^6\), which means multiplying the base, 3, by itself a total of six times. The result, 729, is the final outcome of the evaluation process.

Evaluation confirms our understanding and gives us a tangible result that we can use in further calculations or applications. While calculators or computers can easily compute this, practicing manual evaluation sharpens our mathematical aptitude and deepens our grasp of exponential growth and its implications in various scenarios.