Problem 8 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ \left(4^{-1 / 2}\right)^{-4} $$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Understand Negative Exponents

The expression given is \( (4^{-1/2})^{-4} \). Remember that a negative exponent \( a^{-n} \) means \( \frac{1}{a^n} \). It suggests taking the reciprocal of the base and then applying the positive exponent.

Step 2: Apply the Power of a Power Rule

The expression involves a power raised to another power: \( (4^{-1/2})^{-4} \). We use the rule \( (a^m)^n = a^{m \cdot n} \). Here, \( m = -1/2 \) and \( n = -4 \), so we calculate \( (-1/2) \times (-4) \).

Step 3: Simplify the Exponent

Compute \( (-1/2) \times (-4) = 2 \). So, the expression simplifies to \( 4^2 \).

Step 4: Calculate the Result

Now, calculate \( 4^2 \), which is \( 4 \times 4 = 16 \). Therefore, the expression evaluates to 16.

Key Concepts

Power of a power rule

When you encounter an expression where an exponent is raised to another exponent, such as \((a^m)^n\), use the power of a power rule to simplify it. This rule states that \((a^m)^n = a^{m \cdot n}\). In simpler terms, multiply the exponents. For example, for \((4^{-1/2})^{-4}\), the base is 4, and the exponents are \(-1/2\) and \(-4\). By multiplying these exponents together, we get \((-1/2) \times (-4) = 2\). This simplifies our expression to \(4^2\). The power of a power rule helps by reducing complex exponent chains into simpler calculations.
Always remember:

• Find the exponents involved.
• Multiply them together to find the new exponent.
• Apply the new exponent to the base.

This method is efficient, especially when handling negative exponents, where it streamlines the calculation process.

Simplifying exponents

Simplifying exponents is all about breaking down the expression to its simplest form. When faced with negative exponents, it implies the need for a reciprocal. For instance, an exponent like \(a^{-n}\) is equal to \(\frac{1}{a^n}\). This can seem daunting at first, but it simplifies once steps are followed one at a time.
In our exercise, we started with \(4^{-1/2}\) raised to the \(-4\) power. By applying the power of a power rule, we simplified the exponents to 2 (since \((-1/2) \times (-4) = 2\)). This converts the expression to \(4^2\), which is straightforward to evaluate.

• Identify any negative exponents.
• Use the power of a power rule to simplify.
• Check if the exponent can be reduced further.

By following these steps, an intimidating problem can be transformed into a quick calculation.

Evaluating expressions by hand

Evaluating expressions by hand involves working through mathematical operations step by step without relying on calculators. It's about mentally processing calculations to find the result. In our original problem, \((4^{-1/2})^{-4}\), the steps are simple yet precise when done by hand.
First, understand the role of each exponent and simplify using the power of a power rule. Once you do this, the expression becomes \(4^2\). Calculating \(4^2\) means multiplying 4 by itself: \(4 \times 4 = 16\).
Doing calculations by hand sharpens understanding and builds intuition. Follow these steps:

• Understand the expression and context (e.g., what the exponents denote).
• Simplify the expression using core rules (e.g., power of a power).
• Perform the final calculations manually.

This process may seem longer than using a calculator, but it enriches your grasp of fundamental concepts and improves accuracy over time.