Problem 19 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(\sqrt{16}\) (b) \(\sqrt[4]{16}\) (c) \(\sqrt[4]{\frac{1}{16}}\)

Step-by-Step Solution

Answer

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

Step 1: Evaluating the Square Root

To evaluate \(\sqrt{16}\), we need to find a number that, when multiplied by itself, equals 16. The number 4 satisfies this condition because \(4 \times 4 = 16\). Thus, \(\sqrt{16} = 4\).

Step 2: Evaluating the Fourth Root

Next, we evaluate \(\sqrt[4]{16}\). We are looking for a number that, when multiplied by itself four times, equals 16. In this case, 2 is the number since \(2 \times 2 \times 2 \times 2 = 16\). Therefore, \(\sqrt[4]{16} = 2\).

Step 3: Evaluating the Fourth Root of a Fraction

For \(\sqrt[4]{\frac{1}{16}}\), we need to find a value that when raised to the fourth power equals \(\frac{1}{16}\). Since \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\), \(\sqrt[4]{\frac{1}{16}} = \frac{1}{2}\).

Key Concepts

Square Root

In mathematics, the square root of a number is a value that, when multiplied by itself, results in the original number. It is often represented by the radical symbol \( \sqrt{} \). For example, in the exercise, we are calculating \( \sqrt{16} \). Here, we are searching for a number which, when multiplied by itself, gives us 16. In this case, 4 is the square root of 16 because \( 4 \times 4 = 16 \). Square roots are often used in geometry to determine side lengths of squares from their area and in various fields of science and engineering as well.

You might encounter both positive and negative square roots since both \(4\) and \(-4\) satisfy the condition \( x \times x = 16 \). However, by convention, \( \sqrt{} \) refers to the principal square root, which is always non-negative. The understanding of square roots is foundational to many areas of mathematics.

Fourth Root

The fourth root of a number is a value that, when raised to the power of four, equals the original number. It is often denoted as \( \sqrt[4]{} \). In the problem above, we were tasked with finding \( \sqrt[4]{16} \), which involves figuring out which number multiplied by itself four times results in 16.

By experimentation or familiarity, we find that 2 satisfies this condition, since \(2 \times 2 \times 2 \times 2 = 16\). Thus, \(\sqrt[4]{16} = 2\).

Interestingly, roots like the fourth root are a specific case of an "nth root." The fourth root is commonly used in statistical calculations and can also be useful for computations involving physical properties such as force and resistance.

Fractional Exponents

Fractional exponents provide a compact way to represent roots in mathematical expressions. They allow us to express roots and powers more conveniently within equations. The general formula is \( a^{\frac{1}{n}} \), which represents the nth root of \(a\).

For instance, in the example where we needed to evaluate \(\sqrt[4]{\frac{1}{16}}\), this expression is equivalent to \( \left(\frac{1}{16}\right)^{\frac{1}{4}} \).

Understanding that fractional exponents represent roots can simplify algebraic operations and solve equations where roots are involved. As demonstrated, \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \), so \( \sqrt[4]{\frac{1}{16}}\) reduces to \(\frac{1}{2}\). This unified approach to roots and powers is crucial for advanced mathematics, enhancing both comprehension and computation efficiency.