Problem 3 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression \((2^3)^2\) evaluates to 64.

Step 1: Understand the Expression

The expression \((2^3)^2\) is a power raised to another power. It's essential to recognize this is an exponentiation problem.

Step 2: Apply the Power of a Power Rule

The rule for a power raised to a power is to multiply the exponents. So, for \((a^m)^n\), it simplifies to \(a^{m \cdot n}\).

Step 3: Simplify the Exponent

Apply the rule from Step 2: \((2^3)^2 = 2^{3\cdot2}\). This simplifies to \(2^6\).

Step 4: Calculate the Final Value

Now we need to calculate \(2^6\). This means multiplying 2 by itself 6 times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2\) which equals 64.

Key Concepts

Understanding the Power of a Power Rule

When we encounter an expression like \((a^m)^n\), it's easy to feel overwhelmed if not familiar with the power of a power rule. This mathematical rule is a shortcut to simplify expressions where an exponentiated number is raised to another exponent. Instead of expanding everything, use this rule to multiply the exponents. For example, in the expression \((2^3)^2\), you apply the power of a power rule to get \(2^{3 \times 2}\). This simplifies the problem significantly and turns \((2^3)^2\) into a simpler form \(2^6\).

Here are the key points about the power of a power rule:

• It simplifies exponentiation problems by reducing steps.
• You only need to multiply the exponents while keeping the base the same.
• This rule applies universally to any base and set of exponents.

Techniques for Simplifying Exponents

Simplifying exponents involves reducing complex exponential expressions into simpler ones. When using the power of a power rule, you need to carefully follow each step to ensure accuracy. For instance, in our expression \((2^3)^2\), upon applying the rule, we computed \(2^{3 \cdot 2}\), which simplifies to \(2^6\).

Here’s how to effectively simplify exponents:

• Identify all operations involving exponents in the expression.
• Use appropriate exponent rules, such as power of a power, product rule, and quotient rule.
• Simplify in stages to keep things manageable and avoid errors.

Approaching problems methodically with these techniques streamlines the process and makes dealing with exponents hassle-free.

Evaluating Expressions with Exponents

Once you've simplified an expression involving exponents, the final step is to evaluate it to find the numerical value. After applying the power of a power rule and simplifying to \(2^6\), what remains is the multiplication of 2, six times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Evaluating such expressions involves straightforward arithmetic.

• Compute the repeated multiplication to arrive at the final number.
• In our example, multiplying out gives 64.

Evaluation is the step where the abstract becomes concrete, offering a tangible result to verify the simplification and rule application were done correctly.