Problem 20 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\left(4^{-1 / 2}\right)^{-4}$$

Step-by-Step Solution

Answer

16

Step 1: Understand the Exponent Rules

The problem requires evaluating \( \left( 4^{-1/2} \right)^{-4} \). Remember the rule of exponents that states \((a^m)^n = a^{m \cdot n}\). We will use this rule to simplify the expression.

Step 2: Simplify the Inner Expression

Start with the innermost part of the expression \(4^{-1/2}\). This can be interpreted as the reciprocal of the square root of 4. Thus, \(4^{-1/2} = 1/\sqrt{4} = 1/2\), or equivalently \(2^{-1}\).

Step 3: Apply the Exponent Rule

Now apply the exponent rule to the expression \( (2^{-1})^{-4} \). Use the power of a power property \((a^m)^n = a^{m \cdot n}\) to get \((2^{-1})^{-4} = 2^{-1 \cdot -4} = 2^{4}\).

Step 4: Compute the Final Answer

Calculate \(2^4\). We know that \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). Therefore, the value of the expression is 16.

Key Concepts

Simplifying Radical Expressions

When faced with an expression like \(4^{-1/2}\), it's essential to understand how radical expressions work to simplify them correctly. The term "radical" involves the concept of roots, such as square roots or cube roots.

- The exponent \(-1/2\) can be interpreted in a few steps:
* The negative sign indicates we're dealing with a reciprocal.
* The "1/2" exponent denotes a square root.
* Therefore, \(4^{-1/2}\) represents the reciprocal of the square root of 4.

To simplify the expression, find the square root of 4, which is 2. Taking the reciprocal of 2 gives us \(\frac{1}{2}\). Thus, \(4^{-1/2} = \frac{1}{2}\). Understanding how to break down and interpret each part of a radical expression helps us simplify calculations without a calculator.

Power of a Power Property

The power of a power property is a critical exponent rule used to simplify expressions involving exponents.

- The property states that \((a^m)^n = a^{m \cdot n}\).
* It means raising an exponential expression to another power involves multiplying the exponents.

In our example, we took \((4^{-1/2})^{-4}\) and applied this rule. After simplifying the expression to \(2^{-1}\), applying the power of a power property helps us move forward:
- We calculate \((2^{-1})^{-4}\) as \(2^{-1 \cdot -4}\).
- This simplifies to \(2^4\).

Using this rule ensures expressions involving multiple exponent layers are manageable and correctly evaluated. It's a fundamental concept every math student should master.

Evaluating Expressions Without a Calculator

Often, you might need to evaluate expressions where using a calculator isn't an option. Understanding exponent rules makes this process straightforward and efficient.

- First, break down the expression using well-known rules:
* Use the negative exponent rule to find reciprocals.
* Utilize the power of a power property to simplify exponent expressions.
* Perform straightforward arithmetic operations.

In the given problem, we started with \((4^{-1/2})^{-4}\) and carefully simplified each part using known exponent rules. Finally, we calculated \(2^4\) as \(16\), demonstrating how you can manually compute values step-by-step.

Practicing these techniques builds strong mathematical skills, allowing you to understand the logic behind each step and perform accurate calculations without relying on electronic assistance.