Problem 27 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 1. $$ \left(\frac{1}{4}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

We need to evaluate the expression \(\left(\frac{1}{4}\right)^{1/2}\). This means we need to find the square root of \(\frac{1}{4}\).

Step 2: Apply the Square Root to the Fraction

The square root of a fraction can be found by taking the square root of the numerator and the square root of the denominator separately. Therefore, we take \(\sqrt{1}\) and \(\sqrt{4}\).

Step 3: Calculate the Square Roots

Calculate \(\sqrt{1}\) which is 1, and \(\sqrt{4}\) which is 2. Thus, \(\left(\frac{1}{4}\right)^{1/2} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}\).

Step 4: Simplify the Expression

This gives us the simplified result \(\frac{1}{2}\). Thus, the expression is evaluated to \(\frac{1}{2}\).

Key Concepts

Square Roots

A square root is a number that produces a specified quantity when multiplied by itself. The square root of a number \( x \) is written as \( \sqrt{x} \). For example, the square root of 4 is 2 because \( 2 \times 2 = 4 \). Square roots are fundamental when dealing with powers of 1/2, or expressions where numbers are raised to the exponent of 1/2.

When finding the square root of a fraction, we follow a similar process as with whole numbers. For a fraction \( \frac{a}{b} \), the square root is expressed as \( \frac{\sqrt{a}}{\sqrt{b}} \). This means that we find the square root of the numerator and the square root of the denominator separetly.

• If \( a = 1 \), \( \sqrt{1} = 1 \), because any number multiplied by itself gives 1.
• If \( b = 4 \), \( \sqrt{4} = 2 \), since \( 2 \times 2 = 4 \).

Understanding how to manage square roots, especially with fractions, can simplify many mathematical processes.

Fractions

Fractions are numbers that represent a part of a whole. Each fraction is composed of two numbers: the numerator (top part) and the denominator (bottom part). For example, in \( \frac{1}{4} \), 1 is the numerator and 4 is the denominator.

• The numerator tells us how many parts we have.
• The denominator tells us how many parts make up a whole.

When calculating with fractions, it's essential to follow certain rules. Multiplying or dividing fractions involves working with both the numerators and denominators.

Evaluating expressions with fractions can become easier when breaking down the operations required. Understanding how numerators and denominators interact is crucial. Maintaining balance and order in operations ensures accurate results.

Exponents

Exponents indicate how many times a number, called the base, is multiplied by itself. The notation \( a^n \) means \( a \) multiplied by itself \( n \) times. Exponents are a key component in higher mathematics, simplifying expressions and calculations.

When dealing with exponents that are fractions, such as \( a^{1/2} \), you're finding the square root of \( a \). This involves understanding both exponent laws and square roots simultaneously.

• \( a^{1/2} \) asks for the square root of \( a \).
• \( a^{1} \) is simply \( a \).

Using exponent rules can simplify complex expressions. When evaluating an expression like \( \left( \frac{1}{4} \right)^{1/2} \), recognizing that this is equivalent to finding the square root can guide you through the process smoothly. Understanding this, in conjunction with the principles of fractions, helps solve such exercises efficiently.