Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\(\left(3^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified form of \(\left(3^{3}\right)^{2}\) is 729.

Step 1: Apply the rule of exponents

We start by applying the rule (a^m)^n = a^(m*n), where a is the base, m and n are the exponents. Here, the base 'a' is 3, 'm' is 3, and 'n' is 2. Hence, we rewrite our expression as 3^(3*2).

Step 2: Multiply the exponents

Next, we multiply the exponents. Here we multiply 3 and 2 which gives us 6. So, our expression now becomes 3^6.

Step 3: Evaluate the power

Lastly, we evaluate 3 raised to the power of 6 which gives us the answer 729.

Key Concepts

Power Rule of Exponents

The power rule of exponents is a handy tool when dealing with expressions like \((a^m)^n\). It tells us that we can simplify this by rewriting the expression as \(a^{m \cdot n}\). This means multiplying the two exponents together while keeping the base the same.

When we apply this rule, we make our expressions easier to handle and solve. For example, in the given exercise, the inside exponent is 3, and the outer exponent is 2. By the power rule, we can simplify \((3^3)^2\) to \(3^{3 \times 2}\) which is \(3^6\). This rule is especially useful in algebra, as it helps reduce the problem to a single exponent.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics. It involves rewriting expressions in their simplest form. Through simplification, we can work with smaller numbers and make calculations easier.

In our exercise, the expression \((3^3)^2\) was simplified to \(3^6\) by applying the power rule. This reduced the complexity of the initial expression. It's important to simplify expressions whenever possible, as it helps streamline problem-solving.

• Identify applicable rules (like power rule) for combination of exponents.
• Conduct operations systematically to rewrite expressions.
• Check your work to ensure the simplest form is achieved.

Evaluating Powers

Evaluating powers means computing the value of an expression with exponents. Once simplified, the expression \(3^6\) can be evaluated to find its numerical value. Here, it involves multiplying the base, 3, by itself 6 times.

Steps to evaluate powers:

• Rewrite the expression in a completely simplified form using exponent rules.
• Multiply the base repeatedly according to the exponent, 6 in this case: \(3 \times 3 \times 3 \times 3 \times 3 \times 3\).
• Continue calculating until you reach the final product. In this instance, \(3^6 = 729\).

Assessing powers is often the final step in solving exponent-based problems, providing the answer in a straightforward numerical format.