Problem 30 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 2 \sqrt[3]{24 x^{3} y^{4}}+4 x \sqrt[3]{81 y^{4}} $$

Step-by-Step Solution

Answer

The result is \( 16xy \cdot \sqrt[3]{3y} \).

Step 1: Simplify Each Cube Root Term

First, simplify each cube root separately. For the first term, we have \( \sqrt[3]{24x^3y^4} \). Begin by identifying the perfect cube factors. \( 24 = 8 \times 3 \), so you can write \( 24x^3y^4 \) as \( 8x^3y^3 \times 3y \). The cube root of this expression is \( \sqrt[3]{8x^3y^3 \cdot 3y} = 2xy \cdot \sqrt[3]{3y} \).

Step 2: Simplify the Second Cube Root Term

For the second term, \( \sqrt[3]{81y^4} \), factor \( 81 \) as \( 27 \times 3 \) and express \( y^4 \) as \( y^3 \times y \). Then, rewrite the expression as \( 27y^3 \times 3y \). The cube root can be written as \( \sqrt[3]{27y^3 \cdot 3y} = 3y \cdot \sqrt[3]{3y} \).

Step 3: Substitute Simplified Terms Back Into Original Expression

Replace the cube roots in the original expression with their simplified forms. The expression becomes: \( 2(2xy \cdot \sqrt[3]{3y}) + 4x(3y \cdot \sqrt[3]{3y}) \).

Step 4: Distribute and Combine Like Terms

Distribute the constants in each term: For the first term, this results in \( 4xy \cdot \sqrt[3]{3y} \) and for the second, it is \( 12xy \cdot \sqrt[3]{3y} \). Both terms have common factors, so factor out \( xy \cdot \sqrt[3]{3y} \) from the expression and combine: \( (4 + 12)xy \cdot \sqrt[3]{3y} = 16xy \cdot \sqrt[3]{3y} \).

Key Concepts

Simplifying Radicals

Simplifying radicals is a crucial part of dealing with algebraic expressions, especially when it comes to cube roots like those in our example. A radical expression often involves finding the root of a number or an algebraic term, and simplification is about making that expression as simple as possible.

• First, identify if the radicand (the number under the root) can be factored into smaller, perfect cubes. This helps in simplifying the expression smoothly.
• For instance, with the term \( \sqrt[3]{24x^3y^4} \), we find that 24 can be broken down into \( 8 \times 3 \), effectively finding a perfect cube factor, which is 8 in this case.
• Similarly, identify the powers of the variables, where possible, break them down into perfect cubes too, like \( x^3 \) and \( y^3 \).

This approach of factorization simplifies the radical, turning it into a mixture of whole numbers and simpler cube root expressions, making further calculations easier.

Algebraic Expressions

Understanding algebraic expressions is fundamental in solving many mathematical problems, including the one we are tackling. An algebraic expression can contain constants, variables, and operations (additions, subtractions, multiplications, and divisions).

• In the given problem, you are looking at an expression involving addition of two terms: \( 2 \sqrt[3]{24 x^{3} y^{4}} \) and \( 4 x \sqrt[3]{81 y^{4}} \).
• Each term is a product of constants and variables, paired with cube roots, which means the values are not fixed until the roots are simplified.
• By simplifying the radicals within these terms, you convert complex expressions into more straightforward forms, allowing for easy addition or subtraction.

Simplifying radicals and terms in algebraic expressions help in grasping the overall structure, making complicated problems approachable and solving them less cumbersome.

Real Numbers

In algebra, we often deal with real numbers, which include both rational numbers (like fractions and decimals) and irrational numbers (such as square roots and cube roots that can't be expressed exactly as simple fractions).
This problem assumes all variables and numbers involved are positive real numbers, ensuring that cube roots can be evaluated without delving into imaginary numbers.

• The property of real numbers is essential here since it ensures that expressions like \( \sqrt[3]{24x^3y^4} \) remain valid and solvable.
• Positive real numbers maintain their positivity, which simplifies working with mathematical expressions in many practical contexts.

Understanding this helps in simplifying expressions confidently, knowing no theoretical complexities from imaginary numbers will arise.

Factoring

Factoring is often key to simplifying both numbers and algebraic expressions. By breaking down numbers or expressions into their factors, you reveal simpler components or perfect cubes that aid in further simplification.

• In our exercise, we used factoring to simplify cube roots: \( 24 \) to \( 8 \times 3 \) and \( 81 \) to \( 27 \times 3 \).
• This method also applies to exponents, where a term like \( y^4 \) becomes \( y^3 \times y \), making it easier to take out perfect cubes and simplify the root expression.
• Factoring helps in identifying common terms, allowing for combining like terms more effectively, something we did in the original solution by recognizing common factors in \( 4xy \cdot \sqrt[3]{3y} \) and \( 12xy \cdot \sqrt[3]{3y} \).

Mastering the technique of factoring not only helps simplify complex expressions but also enhances the ability to solve a wide range of algebraic equations.