Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[3]{24} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{24} = 2 \sqrt[3]{3} \)

Step 1: Prime Factorization

First, we need to find the prime factors of the number under the radical, which is 24. The prime factorization of 24 is: \[ 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \]

Step 2: Apply Cube Root to Each Factor

We apply the cube root to each factor of the prime factorization. Since \( \sqrt[3]{x^3} = x \), any cube in the factorization can be simplified:\[ \sqrt[3]{2^3 \times 3} = \sqrt[3]{2^3} \times \sqrt[3]{3} = 2 \times \sqrt[3]{3} \]

Step 3: Combine Simplified Terms

Combine the simplified terms to express the radical in simplest form. The expression becomes:\[ 2\sqrt[3]{3} \]

Key Concepts

Prime Factorization

When dealing with radicals, or roots, one of the most important skills needed is understanding prime factorization. Prime factorization is the process of breaking down a composite number into its prime factors—these are the prime numbers that, when multiplied together, give the original number. For the number 24, the prime factorization is found by dividing 24 by the smallest prime number, which is 2. We continue dividing until we can no longer divide evenly.

• Divide 24 by 2 to get 12.
• Divide 12 by 2 to get 6.
• Divide 6 by 2 to get 3, which is a prime number.

This process shows that 24 is composed of the factors: \[24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3\]Understanding this allows us to simplify radicals effectively. Knowing how to prime factor a number assists with isolating cubes, squares, or higher powers that can be further simplified.

Cube Roots

A cube root asks the question: What number multiplied by itself three times equals the number under the radical? In algebraic terms, for a number \( x \), the cube root is represented as \( \sqrt[3]{x} \). Cube roots are similar to square roots but with a cube root, we search for groups of three, rather than two.When we apply a cube root to the expression \( \sqrt[3]{24} \), it is essential first to express 24 in its prime factorized form, which is \( 2^3 \times 3 \). * Each set of three identical factors (like \( 2^3 \)) can be 'extracted' from under the radical. This means that \( \sqrt[3]{2^3} = 2 \) because 2 multiplied by itself three times equals 8, and \( 8 = 2^3 \). The remaining factor, 3, stays under the cube root, resulting in the simplified expression \( 2 \cdot \sqrt[3]{3} \). Understanding how to take cube roots is key in simplifying radicals to their simplest form, which is particularly useful in algebra.

Radicals in Algebra

Radicals often appear in algebraic expressions, and knowing how to simplify them is crucial. The expressions \( \sqrt{ } \) or \( \sqrt[n]{ } \) signify that a root is present in an expression. For radicals to be in their simplest form, any perfect powers matching the root within the radicand (the number under the root) should be extracted.In algebra, handling radicals involves more than just roots of numbers; it often includes variables. An expression like \( \sqrt[3]{x^3} \) simplifies neatly to \( x \) because the cube root 'cancels' the cube of \( x \). However, not all components within the radical can always be simplified and may remain as part of the expression.
For instance, in the problem \( \sqrt[3]{24} \), the simplified form \( 2\sqrt[3]{3} \) is achieved because the \( 2^3 \) part is perfect, while the 3 isn't - leaving it under the radical. This knowledge enables managing radicals effectively, making you more comforted with algebraic equations and expressions.