Problem 15 - Algebra and Trigonometry

Question

\(15-24\) . Evaluate each expression. $$ \begin{array}{llll}{\text { (a) } \sqrt{16}} & {\text { (b) } \sqrt[4]{16}} & {\text { (c) } \sqrt[4]{\frac{1}{16}}}\end{array} $$

Step-by-Step Solution

Answer

(a) 4, (b) 2, (c) \( \frac{1}{2} \)

Step 1: Evaluate \( \sqrt{16} \)

To solve \( \sqrt{16} \), we need to find a number that when multiplied by itself gives 16. The number \( 4 \times 4 = 16 \), so \( \sqrt{16} = 4 \).

Step 2: Evaluate \( \sqrt[4]{16} \)

To solve \( \sqrt[4]{16} \), we need to find a number that when multiplied by itself four times gives 16. The number \( 2 \times 2 \times 2 \times 2 = 16 \), so \( \sqrt[4]{16} = 2 \).

Step 3: Evaluate \( \sqrt[4]{\frac{1}{16}} \)

To solve \( \sqrt[4]{\frac{1}{16}} \), we need to find the fourth root of \( \frac{1}{16} \). This is equivalent to \( \left(\sqrt[4]{1}\right) / \left(\sqrt[4]{16}\right) \). From the previous step, we know \( \sqrt[4]{16} = 2 \), and \( \sqrt[4]{1} = 1 \) because any root of 1 is 1. So, \( \sqrt[4]{\frac{1}{16}} = \frac{1}{2} \).

Key Concepts

Square Root

A square root is essentially the inverse operation of squaring a number. To find the square root of a given number, you're looking for a value that, when multiplied by itself, produces the original number. Consider the square root symbol \( \sqrt{} \) as the guide to discovering that number.
A helpful way to think of square roots is to view them as a pair of identical factors. For instance, when evaluating \( \sqrt{16} \), you're looking for a number that when squared equals 16. The answer here is 4, since \( 4 \times 4 = 16 \).

• If the number is a perfect square, like 16, the result is a whole number.
• Not all numbers are perfect squares. For non-perfect squares, the result will be an irrational number, like \( \sqrt{2} \).

Understanding square roots helps simplify expressions and solve equations effectively. With practice, finding square roots becomes second nature.

Fourth Root

The fourth root extends the concept of square roots further. Instead of finding a number that, squared, results in the original, for fourth roots, you look for a number that raised to the power of four gives the original number.
Using the fourth root, represented by \( \sqrt[4]{} \), let's evaluate \( \sqrt[4]{16} \). In this case, you're searching for a number that, when multiplied by itself four times, equals 16. The answer is 2 because \( 2 \times 2 \times 2 \times 2 = 16 \).

• The fourth root of 1, as in \( \sqrt[4]{1} \), is 1. Any root of 1 will always be 1.
• Finding fourth roots is useful in algebra when dealing with powers and polynomial equations.

Just like the square root, the fourth root reveals fundamental insights into the nature of numbers and their relationships.

Simplifying Radicals

Simplifying radicals involves reducing them to their simplest form. This means breaking them down into more manageable numbers, where possible.
To simplify involves repetitively breaking down a radical until no further simplification is possible. Consider \( \sqrt[4]{\frac{1}{16}} \). Here, it's equivalent to \( \frac{\sqrt[4]{1}}{\sqrt[4]{16}} \). We already know that the fourth root of 16 is 2, and the fourth root of 1 is 1. So, \( \sqrt[4]{\frac{1}{16}} = \frac{1}{2} \).
Simplifying radicals helps:

• Make equations easier to solve.
• Provide clearer insights into problems involving roots and exponents.

Reducing a radical to its simplest form often requires knowledge of prime factors and understanding of root principles. Practice and familiarity with these concepts enable easier problem solving and algebraic manipulation.