Problem 35 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[3]{405 x^{12} y^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^4y\sqrt[3]{3y}\).

Step 1: Factor the Radicand

Factor the expression inside the cube root. The number 405 can be factored into prime factors: 405 = 3^4 * 5. Thus, the expression becomes \(405 = 3^4 \times 5 \), so now, \( \sqrt[3]{405 x^{12} y^{4}} = \sqrt[3]{3^4 \times 5 \times x^{12} \times y^4}\).

Step 2: Apply Cube Root to Each Factor

The cube root can be applied to each individual factor: \( \sqrt[3]{3^4 \times 5 \times x^{12} \times y^4} = \sqrt[3]{3^4} \times \sqrt[3]{5} \times \sqrt[3]{x^{12}} \times \sqrt[3]{y^4}\).

Step 3: Simplify Each Factor

Simplify each cube root: - \(\sqrt[3]{3^4} = 3^{4/3} = 3^1 \times 3^{1/3}\), so it simplifies to \(3 \sqrt[3]{3} \);- \(\sqrt[3]{x^{12}} = x^{12/3} = x^4\);- \(\sqrt[3]{y^4} = y^{4/3} = y \cdot y^{1/3} \).Combine to get the simplified expression: \(3x^4y \cdot \sqrt[3]{3y}\).

Step 4: Combine and Simplify Further if Necessary

Combine the simplified terms to express the final simplified radical expression as: \(3x^4y\sqrt[3]{3y}\).

Key Concepts

Cube Root Simplification

Cube root simplification involves simplifying expressions that have a cube root. A cube root is essentially the opposite of raising a number to the third power, or cubing it. When you simplify a cube root, you're trying to express it in its simplest form. Let's break it down using an example with numbers and variables.

Imagine you have an expression like \( \sqrt[3]{a} \). When you simplify this, you're finding a number or expression that, when used three times in a multiplication, gives you \( a \). This process might involve working with prime factors or other algebraic manipulations.

In the cube root of a number like \( \sqrt[3]{405} \), you'd look for factors you can "pull out" of the cube root by expressing 405 in terms of its prime factors, such as \( 3^4 \times 5 \). For cube root simplification,

• Take out factors raised to powers that are multiples of three because they simplify perfectly under a cube root.
• Leave inside any factors that you can't simplify completely.

Prime Factorization

Prime factorization is a technique used to break down a complex number into a product of prime numbers. Prime numbers are numbers that only have two divisors: 1 and themselves. They are the building blocks of all numbers.

To perform prime factorization, you divide the number by the smallest prime number possible (starting with 2) and keep dividing until all you have left are prime numbers. It's like peeling the layers of an onion, except every layer is a prime number.

• For example, the number 405 can be factored into \(3^4 \times 5\).
• Here, 3 is a prime number, and so is 5. 405 = 81 × 5 = (3×3×3×3)×5.

Prime factorization is essential for simplifying radical expressions because it allows you to identify and extract factors that can come out of the radical, especially when the root involves a power like a cube root.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. When dealing with algebraic expressions within radical expressions, it's crucial to apply algebraic operations properly to simplify them.

In our example, we have \( \sqrt[3]{405 x^{12} y^{4}} \). Here:

• \(x^{12}\) indicates that x is raised to the twelfth power, and a similar notation is used for y.
• Simplifying the cube root of \(x^{12}\), for instance, involves dividing the exponent by 3, as the cube root "undoes" cubing it.

To simplify radical expressions involving algebraic terms:

• Divide the powers by the root's degree (in this case, 3).
• Write down the simplified base with the reduced exponent.
• If some parts remain under the root, as with \( y^{4} \), simplify what can be taken out and leave the rest inside.

Handling these with care can help convert complex expressions into simpler, more understandable forms.