Problem 20 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[4]{16} $$

Step-by-Step Solution

Answer

The fourth root of 16 is 2.

Step 1: Understand the Expression

The expression given is \(\sqrt[4]{16}\). This indicates that we need to find the fourth root of 16, which means finding a number that, when raised to the power of 4, gives 16.

Step 2: Solve the Equation

To find \(\sqrt[4]{16}\), we need to solve \(x^4 = 16\). We will determine which real number raised to the fourth power results in 16.

Step 3: Find the Fourth Root

Consider \(x = 2\). Check if \(2^4 = 16\). Indeed, by calculating, we see that \(2 \times 2 \times 2 \times 2 = 16\). So, \(2\) is a potential fourth root.

Step 4: Verify the Solution

Confirm that the only positive real fourth root is indeed 2. Given that we are looking in the set of real numbers, the positive fourth root is the relevant answer here.

Key Concepts

Understanding Real Numbers

Real numbers form a complete numeric system used in everyday life and advanced mathematics. They include all the numbers that you may already know, including:

• Whole numbers: 0, 1, 2, 3, ...
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Fractions or rational numbers: 1/2, 3/4, 5/8, ...
• Irrational numbers: \(\pi, \sqrt{2},\) etc.

Real numbers are any point along the number line, meaning they can be either positive, negative, or zero. It is a rich set that allows us to perform a wide variety of calculations. In the context of our exercise, we focus on finding a real number which, when raised to a given power, equals another real number. Understanding this helps us find solutions in problems involving roots and exponents, like identifying the fourth root of 16.

What Are Radicals?

Radicals are symbols used to signify taking roots of numbers. You often encounter radicals in the form of a square root or cube root, but they can represent any degree of root, such as a fourth root or fifth root.
When we see the symbol \(\sqrt{a}\), it indicates finding a number which, when squared, equals \(a\). Similarly, \(\sqrt[3]{a}\) is the cube root of \(a\), meaning we need a number that, when cubed, gives \(a\). For our exercise, \(\sqrt[4]{16}\) means we're looking for a number that, when raised to the fourth power, equals 16.

• The radical symbol \(\sqrt{}\) always has an index, which tells us which root we need to find (like 4 in fourth root).
• Understanding radicals is crucial for simplifying expressions and solving equations that involve roots.

In practice, radicals can make complex roots simple by providing a clear visual representation of the operation we're aiming to perform.

Exploring Rational Exponents

Rational exponents are a way of expressing roots using exponents. Instead of writing \(\sqrt[4]{16}\) as a radical, we can express it as a number raised to a fractional power: \(16^{1/4}\). This concept combines the ideas of exponents and radicals in a compact form.
Here's how it works:

• Any number raised to the power of \(\frac{1}{n}\) equals the \(n\)-th root of that number.
• For example, \(16^{1/4}\) is the fourth root of 16, the same as \(\sqrt[4]{16}\).
• Equations involving rational exponents can be solved in a similar way to regular exponent equations.

Rational exponents allow us to use the rules of exponents seamlessly, enhancing our ability to handle complex equations. They provide a powerful method to simplify expressions and understand roots in a more algebraic way.