Problem 62 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[4]{(-2)^{4}}$$

Step-by-Step Solution

Answer

The evaluated expression \(\sqrt[4]{(-2)^{4}}\) is equal to 2

Step 1: Square the number

Let's begin by squaring the number -2, which gives us \( (-2)^{4} = 16 \)

Step 2: Find the fourth root

Once obtained the squared value, we will find the fourth root of the result, that is \(\sqrt[4]{16} = 2\)

Key Concepts

Real Numbers

Real numbers include all rational and irrational numbers. They can be positive, negative, or zero. Rational numbers can be expressed as fractions, such as 3/4 or 7. Irrational numbers are those like \( \pi \) or \( \sqrt{2} \), which cannot be written as exact fractions.

• All real numbers lie on the number line, creating a continuum of values.
• Real numbers are vital in everyday math, helping us measure and calculate real-world quantities.

In our exercise, -2 and 16 are examples of real numbers. The operations performed do not break the real number boundaries, as there is no division by zero or root of a negative number (when doing even roots). This ensures the solutions remain within the realm of real numbers.

Roots and Radicals

Roots and radicals are ways to describe numbers that when multiplied by themselves a specific number of times result in a given value.

• The square root \( \sqrt{} \) is one of the most common, followed by the cube root \( \sqrt[3]{} \), and so on.
• An \( n \)-th root is written as \( \sqrt[n]{x} \), and it asks what number, multiplied by itself \( n \) times, equals \( x \).

In our original exercise, we deal with the fourth root of a number. Finding the fourth root of 16, \( \sqrt[4]{16} \), means finding a number that when raised to the 4th power equals 16. Here, that number is 2.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number.

• For example, \( (-2)^{4} \) means multiplying -2 by itself four times: \( (-2) \times (-2) \times (-2) \times (-2) \).
• The outcome of \( (-2)^{4} \) is 16, as the negative sign disappears because every multiplication of two negative numbers results in a positive.

Understanding exponents helps in breaking down complex expressions and solving them step by step, just as we saw in the exercise, where squaring -2 four times gave us a positive 16, a pivotal step in calculating the fourth root. This demonstrates how exponents allow us to work efficiently with large numbers and roots.