Problem 28 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 1. $$ \left(\frac{1}{16}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The value of \( \left(\frac{1}{16}\right)^{1 / 2} \) is \( \frac{1}{4} \).

Step 1: Understand the Expression

The expression we are trying to evaluate is \( \left(\frac{1}{16}\right)^{1 / 2} \). This expression is indicating that we need to find the square root of the fraction \( \frac{1}{16} \).

Step 2: Simplify the Expression

Taking the square root of a fraction is equivalent to taking the square root of the numerator and the denominator separately. Thus, \( \left(\frac{1}{16}\right)^{1/2} = \left(\frac{1^{1/2}}{16^{1/2}}\right) \).

Step 3: Calculate the Square Roots

The square root of 1 is 1, and the square root of 16 is 4. Therefore, \( \left(\frac{1^{1/2}}{16^{1/2}}\right) = \frac{1}{4} \).

Step 4: Write the Final Answer

The simplified value of the expression \( \left(\frac{1}{16}\right)^{1 / 2} \) is \( \frac{1}{4} \).

Key Concepts

square root

The term "square root" refers to a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 times 4 equals 16. This operation is often denoted by the radical symbol \( \sqrt{} \). Another notation used for square root, especially in expressions, is with exponents. When you see \( x^{1/2} \), it indicates the square root of \( x \).
This concept becomes very helpful when working with fractions or when simplifying expressions in algebra. Remember:

• The square root of 1 is always 1 since 1 times 1 is 1.
• The square root of perfect squares like 4, 9, 16, etc., resolve to whole numbers.

Understanding square roots is crucial in evaluating expressions like \( \left(\frac{1}{16}\right)^{1/2} \). By interpreting this expression as finding the square root of \( \frac{1}{16} \), we can systematically arrive at the solution.

fractions

Fractions represent a part of a whole and are composed of a numerator (the top number) and a denominator (the bottom number). Simplifying fractions involves reducing them to their simplest form, such that the numerator and denominator have no common factors other than 1.
To work with fractions when evaluating expressions like exponents or roots, you usually apply operations separately on the numerator and denominator. For instance:

• If you're finding the square root of a fraction, apply the square root to both the numerator and the denominator separately.
• \( \left(\frac{1}{16}\right)^{1/2} \) becomes \( \frac{1^{1/2}}{16^{1/2}} \).

When dealing with fractional exponents or square roots, grasping how fractions interact within the expression helps demystify and solve it accurately. Always check to ensure fractions are fully simplified after processing.

simplifying expressions

Simplifying expressions often means making an expression as basic as possible without changing its value. This could involve reducing fractions, distributing terms, or, as seen in our example, evaluating exponents.
A key part of simplifying is breaking down an expression systematically. Consider:

• Finding square roots of easy numbers to simplify exponentiated terms.
• Simplifying \( \frac{1}{16} \) by expressing it as \( \frac{1^{1/2}}{16^{1/2}} \) and taking the square root.

The simplest form of an expression allows for clearer understanding and quicker calculations. Whether working with radicals, fractional exponents, or just arithmetic operations, aim to reduce the expression comprehensively.