Problem 15 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

So, the simplification of expression \( \left(2^{3} \cdot 3^{2}\right)^{2} \) is 5184.

Step 1: Apply the Power Rule

To solve the expression, apply the power rule which states that \( (a^{m})^{n} = a^{m*n} \). Applying this rule to \( \left(2^{3} \cdot 3^{2}\right)^{2} \) gives \( 2^{3*2} \cdot 3^{2*2} = 2^{6} \cdot 3^{4} \).

Step 2: Evaluate Each Term

Next, evaluate each term to its simplest form. The value of \(2^6\) is 64 and \(3^4\) is 81.

Step 3: Simplify Further

Once each term is evaluated, simplify further. Thus, \(2^{6} \cdot 3^{4} = 64 \cdot 81 = 5184 \).

Key Concepts

Power Rule

The power rule is a vital concept in exponentiation. It helps you easily manage expressions involving exponents raised to another power. Imagine you have an expression like \((a^m)^n\), where a is any number, m is the initial exponent, and n is the power to which it's being raised again. The power rule comes in handy by simplifying this complex setup to just \(a^{m \times n}\). This means you multiply the original exponent by the new one instead of calculating it step-by-step repeatedly.

In our example, \((2^{3} \cdot 3^{2})^{2}\), we can break it down into smaller parts by applying the power rule separately to each base:

• For base 2, it's \((2^3)^2 = 2^{3 \times 2} = 2^6\).
• For base 3, it's \((3^2)^2 = 3^{2 \times 2} = 3^4\).

This simplification makes further calculations much easier and helps in understanding how to handle larger exponent-related problems.

Simplifying Expressions

Simplifying expressions is about making a math problem easier to work with. Once you have used the power rule to adjust your exponents, the next step is to simplify the mathematical expression itself. This process involves evaluating each part of the expression to find their numerical values.

For our expression \(2^6 \cdot 3^4\), the simplification process involves calculating:

• \(2^6 = 64\).
• \(3^4 = 81\).

By finding these values, you're reducing the complexity of the expression, turning powers into manageable numbers. Simplifying an expression doesn't necessarily mean you can only play with fractions—it means making any expression easier to work with. Later, you'll use these outcomes to continue solving the equation.

Evaluating Expressions

After you've applied the power rule and simplified the expression, evaluation leads you to the final answer. Evaluating an expression means performing the arithmetic operations as instructed to reach the solution. It's like solving the final puzzle.

In the case of our example \(2^6 \cdot 3^4\), the evaluation step involves straightforward multiplication:

• Multiply the simplified results, 64 (from \(2^6\)) and 81 (from \(3^4\)).
• This gives us \(64 \times 81 = 5184\).

Thus, evaluating expressions leads to obtaining a single, final number. It's crucial to follow through with clean operations to avoid errors. Understanding each step clearly ensures you know how to tackle similar problems in the future with confidence.