Problem 12 - 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. $$ \sqrt[3]{16} $$

Step-by-Step Solution

Answer

\( 2 \cdot \sqrt[3]{2} \)

Step 1: Express the Radicand as a Product of Primes

First, we express the number 16 as a product of prime numbers. The number 16 can be written as \( 2^4 \).

Step 2: Apply the Cube Root to the Prime Factorization

Since we are dealing with a cube root, we can apply the radical to the prime factorization: \( \sqrt[3]{16} = \sqrt[3]{2^4} \).

Step 3: Simplify Using Properties of Radicals

We can simplify \( \sqrt[3]{2^4} \) by separating it into \( \sqrt[3]{2^3} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2} \). We do this because \( \sqrt[3]{2^3} \) simplifies to 2, as taking the cube root of a cube gives the base number.

Key Concepts

Cube Roots

Cube roots are a unique type of radical expression. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is represented with the radical symbol having an index of 3. For instance, \[ \sqrt[3]{x} \]is the cube root of \(x\). This means we need to find a number which, when cubed (or raised to the power of three), equals \(x\).
Cube roots provide important value in simplifying problems involving volumes or larger power simplifications. Unlike square roots, cube roots can yield negative values if the radicand (the number under the radical) is negative. This is because multiplying a negative number three times results in a negative product. It's essential to understand the nature of cube roots in simplifying calculations or equations involving such roots.
Let's explore further by discussing prime factorization, a key technique in simplifying radicals.

Prime Factorization

Prime factorization is an essential step in simplifying radicals, especially with cube roots. It involves expressing a number as a product of its prime factors—which are numbers only divisible by 1 and themselves, such as 2, 3, 5, 7, etc. The process helps to break down complex radicands into simpler components.
For instance, the number 16 can be expressed through its prime factors as:

• 16 is divisible by 2 (since it's even) which gives: 16 = 2 × 8
• 8 is again divisible by 2, hence: 8 = 2 × 4
• Finally, 4 is divisible by 2 yielding: 4 = 2 × 2

This simplifies 16 as \(2^4\).
Prime factorization is a systematic method that greatly aids in applying the properties of radicals effectively. Once broken down, we can utilize these prime components to simplify the cube root by looking for triples of the same number, as demonstrated in the solution steps.

Properties of Radicals

Understanding properties of radicals is vital for simplifying expressions like cube roots. These properties allow manipulation of radical expressions for easier computation:

• Product Property: States that the nth root of a product is the product of the nth roots. That is: \[ \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \]
• Power Property: Suggests that the nth root of \(a^m\) is \[ a^{m/n} \] provided that all roots are real.
• Simplification: If a complete power that matches the radical index can be extracted from an expression, it simplifies the expression. For cube roots, \( \sqrt[3]{2^3} \) directly simplifies to 2.

In the solution, applying these properties allows us to break down \[ \sqrt[3]{2^4} \] into simpler forms of \[ \sqrt[3]{2^3} \cdot \sqrt[3]{2} \]which ultimately simplifies to 2 times\( \sqrt[3]{2} \).
Mastering these radical properties ensures effective simplification and empowers solving various algebraic expressions with multiple radical components efficiently.