Problem 55 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) 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: Understand the root type

Our root here is a cubic root which means we are looking for a number, which when multiplied by itself thrice, will give the number under the root, which happens to be 125 in this case.

Step 2: Evaluate the cubic root

The cube root of 125 can be written as \( \sqrt[3]{125} = 5 \), because when 5 is multiplied by itself thrice (5 * 5 * 5), we get 125. So the cube root of 125 is 5.

Key Concepts

Cubic Root Evaluation

Understanding how to evaluate cubic roots is fundamental in algebra. The cubic root of a number is a value that, when multiplied by itself three times, yields the original number. This concept is symbolically represented by \( \sqrt[3]{x} \) where x is the number we wish to find the cubic root of.

For example, in the exercise \( \sqrt[3]{125} \) we're looking for a number that when cubed (multiplied by itself three times), equals 125. Through either memorization of small perfect cubes or simple trial and error, we can determine that this number is 5, as \( 5 \times 5 \times 5 = 125 \). Thus, the cubic root of 125 is 5. When approaching cubic root evaluation, remember that it's essentially asking 'What number, when used as a factor three times, gives the specified result?'

Radicals in Algebra

Radicals are symbols used in algebra to indicate roots, including square roots, cubic roots, and higher-order roots. In general, the radical symbol with a little number called the index (\( n \) in \( \sqrt[n]{x} \) indicates the n-th root of \( x \). When the index is 3, as in our exercise \( \sqrt[3]{125} \) it’s called a 'cubic root'.

Understanding how radicals work is crucial for solving algebraic problems. It requires familiarity with a few properties, such as the product and quotient of radicals, and dealing with both rational and irrational numbers. For instance, \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \) and \( \sqrt[3]{a^3} = a \) when \( a \) is positive. It's important to recognize that not all radical expressions result in integers and can remain as radicals if they cannot be simplified to rational numbers.

Exponents and Roots

Exponents and roots are closely related concepts within algebra; they are essentially mathematical 'opposites'. An exponent indicates how many times a number, known as the base, is multiplied by itself. A root, on the other hand, tries to reverse this process. For instance, an exponent of 2 (squared) and a square root are inverse operations — \( x^2 \) and \( \sqrt{x} \) undo each other.

Similarly, a cubic exponent (to the power of 3) and a cubic root are inverses. If you have a number \( x \) and you apply a cube operation \( x^3 \) and then a cubic root \( \sqrt[3]{x^3} \) , you will return to the original number, assuming \( x \) is positive. Knowing this, exponents and roots can be used to solve for unknowns in algebraic equations, simplify expressions, and understand the scaling of quantities in real-world contexts. They're vital tools that allow mathematicians to work with powers and perform operations that are otherwise not straightforward.