Problem 49 - College Algebra

Question

Evaluate each expression in Exercises \(49-60\), or indicate that the root is not a real number. $$\sqrt[3]{125}$$

Step-by-Step Solution

Answer

The cube root of 125 is 5.

Step 1: Identify the Cube Root

The first step is to identify the cube root. The cube root is written as \(\sqrt[3]{125}\). Here, 125 is the radicand, which is the number we are taking the cube root of.

Step 2: Calculate the Cube Root

The next step is to calculate the cube root. We are looking for a number that when cubed (multiplied by itself and then by itself again) gives us 125. By trying different numbers, we find that when 5 is cubed, we get 125. Thus, \(5^3 = 125\).

Step 3: Verify the Result

After finding the cube root, we must verify our result. We can do this by cubing the result. So, \(5^3 = 5*5*5 = 125\), which verifies that our cube root calculation is correct.

Key Concepts

Radicand

When delving into the realm of roots and exponents, one comes across the term radicand. A radicand is the number within the radical sign that you are finding the root of. In the expression \( \sqrt[3]{125} \), the radicand is 125. It is the 'subject' of the root operation.

Familiarizing with the term radicand is fundamental, as many of the properties of roots will relate back to this number. For instance, when you're instructed to simplify a root, it's the radicand you often manipulate. If the radicand is a perfect cube in this case, such as 125, we can find a whole number solution representing the cube root. Understanding this concept is a crucial stepping stone to maneuvering through more complex problems in algebra and higher mathematics.

Exponentiation

In mathematics, the concept of exponentiation is a form of shorthand for repeated multiplication. The expression \( 5^3 \) represents 5 multiplied by itself three times (5*5*5). The number 5 is the base, and the number 3 is the exponent, which tells us how many times to multiply the base by itself. In the cube root example \( \sqrt[3]{125} \), we reverse this process to find the base.

Exponents aren't limited to whole numbers or integers. They can represent a wide variety of operations including square roots, cube roots, and more generally, nth roots. Exponentiation is a critical operation in algebra, calculus, and other areas of mathematics, as well as in fields that rely on mathematics, such as engineering, physics, and computer science. Developing a solid understanding of how exponentiation works enables students to solve a wide range of problems effectively.

Real Numbers

The set of real numbers is a fundamental building block in mathematics, encompassing both rational numbers (like 6 or \( \frac{1}{2} \)) and irrational numbers (like \sqrt{2} or \( \pi \)). Real numbers can be plotted on a number line, which visually represents their size and order.

When you're working with cube roots, such as \( \sqrt[3]{125} \), the result is often a real number. However, it’s possible for cube root operations to result in non-real numbers when performed within the realm of complex numbers. Nonetheless, for all positive real numbers, there's always a positive real number that is its cube root. Recognizing if a root will yield a real number is an important part of solving these types of problems. The cube root of 125 results in the real number 5, which fits neatly into this system.