Problem 33 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 4^{-1 / 2} $$

Step-by-Step Solution

Answer

\(4^{-1/2} = \frac{1}{2}\)

Step 1: Understand the Negative Exponent

The expression given is \(4^{-1/2}\). A negative exponent implies that we need to take the reciprocal of the base raised to the positive of that exponent. Therefore, \(4^{-1/2} = \frac{1}{4^{1/2}}\).

Step 2: Evaluate the Fractional Exponent

The positive fractional exponent \(1/2\) indicates a square root. Therefore, \(4^{1/2}\) is the square root of 4. Since the square root of 4 is 2, we have \(4^{1/2} = 2\).

Step 3: Apply the Reciprocal Operation

Since \(4^{-1/2} = \frac{1}{4^{1/2}}\) and \(4^{1/2} = 2\), we substitute to find \(\frac{1}{2}\).

Key Concepts

Fractional Exponents

Fractional exponents may seem daunting at first, but they are just a way to represent roots in a more flexible form. When you see an exponent like \( \frac{1}{2} \), this indicates a square root. More generally, an exponent of \( \frac{1}{n} \) corresponds to the nth root of a number. This fractional exponent makes complex mathematical calculations much simpler once you understand the basics.When evaluating expressions with fractional exponents, it often helps to rewrite these exponents as roots. For example, if you have an expression like \( 4^{\frac{1}{2}} \), it's equivalent to saying "the square root of 4." This understanding allows you to not just compute values more easily, but also to gain a deeper insight into the relationships between powers and roots.

Reciprocal

The concept of reciprocal is fundamental in mathematics and involves flipping a number. In the context of exponents, a negative exponent like \( -\frac{1}{2} \) tells us to take the reciprocal before proceeding with any other operation. The reciprocal of a number \( x \) is calculated as \( \frac{1}{x} \). For instance, when you have \( 4^{-\frac{1}{2}} \), it suggests that you're taking the reciprocal of \( 4^{\frac{1}{2}} \). So, once you calculate \( 4^{\frac{1}{2}} \), you transform it by placing it in the denominator, essentially giving you \( \frac{1}{2} \).

Square Root

Square roots are one of the most familiar types of fractional exponents. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). In the given problem, \( 4^{\frac{1}{2}} \) represents the square root of 4. Evaluating it reveals that the square root of 4 is 2 since \( 2 \times 2 = 4 \).Understanding square roots allows you to evaluate expressions like these quickly. They are a building block for more comprehensive mathematical studies and serve as the foundation for operations involving rational exponents.

Evaluate Expressions Without a Calculator

Evaluating mathematical expressions without a calculator emphasizes understanding the underlying principles rather than relying on technology. To evaluate expressions such as \( 4^{-\frac{1}{2}} \) manually:

• Identify that a negative exponent requires finding the reciprocal.
• Recognize that the fractional exponent represents a root.
• Use these insights to break down the calculations step by step.

For instance, by recognizing that \( 4^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}} \) and knowing \( 4^{\frac{1}{2}} = 2 \), you can determine that the expression equals \( \frac{1}{2} \).This methodical approach improves problem-solving skills and enhances your mathematical intuition, allowing you to tackle even more challenging problems.