Problem 33 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The value is \(\frac{1}{2}\).

Step 1: Understand the Negative Exponent

A negative exponent indicates that we take the reciprocal of the base raised to the positive of the given exponent. Therefore, \(4^{-1/2} = rac{1}{4^{1/2}}\).

Step 2: Simplify the Exponent \(1/2\)

An exponent of \(1/2\) means "square root." Thus, \(4^{1/2}\) is equivalent to \( ext{the square root of 4}\).

Step 3: Calculate the Square Root of 4

The square root of 4 is 2 because \(2 imes 2 = 4\). Therefore, \(4^{1/2} = 2\).

Step 4: Apply the Reciprocal

Since \(4^{-1/2} = rac{1}{4^{1/2}}\), and we know \(4^{1/2} = 2\), the expression simplifies to \(rac{1}{2}\).

Key Concepts

Reciprocal

The concept of a reciprocal is essential in understanding negative exponents. When we talk about reciprocals, we mean that we invert a number or an expression to its reciprocal form. For instance, the reciprocal of a number is simply one divided by that number. If we take a number, say `x`, the reciprocal is expressed as \( \frac{1}{x} \). This concept is very useful in addressing expressions with negative exponents. - A negative exponent, such as in the expression \( 4^{-1/2} \), indicates that we take the reciprocal of the base, \(4\), raised to the corresponding positive exponent (which is \( \frac{1}{2} \)).- Thus, \( 4^{-1/2} \) transforms into \( \frac{1}{4^{1/2}} \).This idea of taking reciprocals is key to simplifying expressions with negative exponents, making the complex expressions more digestible. By flipping the base as a reciprocal, you effectively "turn" the negative exponent into its positive counterpart by working with the inverse.

Square Root

The square root is a specific type of exponentiation expressed as raising to the power of \( \frac{1}{2} \). Understanding square roots helps in simplifying expressions like \( 4^{-1/2} \). The term square root denotes a value that, when multiplied by itself, returns the original number. - For example, the square root of 4 is 2 because \( 2 \times 2 = 4 \).- In mathematical notation, this is expressed as \( 4^{1/2} = \sqrt{4} = 2 \).By recognizing \( 4^{1/2} \) as a square root, we simplify the process of evaluating expressions involving roots. Using the square root, we can reduce the expression \( 4^{-1/2} \) to \( \frac{1}{2} \). This step simplifies working with powers and roots, making calculations manageable and clear.

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising numbers to powers. This concept is pivotal in understanding expressions like \( 4^{-1/2} \). Exponentiation dictates how many times a base number is multiplied by itself.- A positive exponent, such as \( n \), denotes that a number is multiplied by itself \( n \) times.- A negative exponent, however, switches the approach by indicating the reciprocal of the base raised to the absolute value of the exponent.For example, \( 4^{-1/2} \) involves both a negative exponent and a fractional exponent: - The negative sign directs us to take a reciprocal. - The fraction \( \frac{1}{2} \) denotes a square root.This dual aspect requires recognizing both the inverse operation and the root extraction. Exponentiation ties together these operations, turning complex expressions into manageable calculations by breaking them down into these foundational concepts.