Problem 30 - Intermediate Algebra

Question

Add or subtract. $$ 2 \sqrt[3]{24 x^{3} y^{4}}+4 x \sqrt[3]{81 y^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4xy \sqrt[3]{3y} + 12xy \sqrt[3]{3}\).

Step 1: Simplify the Radicals

First, simplify the cube roots in each term.For the first term:\[2 \sqrt[3]{24x^3y^4}\]We break down each component inside the cube root:- Cube root of 24 involves prime factorization: \(24 = 2^3 \times 3\)- Cube root of \(x^3\) is \(x\)- Cube root of \(y^4 = y^3 \times y\); cube root of \(y^3\) is \(y\)Thus, the simplification of the first term becomes:\[2 \times 2 \times xy \sqrt[3]{3y} = 4xy \sqrt[3]{3y}\]For the second term:\[4x \sqrt[3]{81y^4}\]We break down each component inside the cube root:- Cube root of 81 involves \(81 = 3^4 = 3^3 \times 3\)- Cube root of \(y^4 = y^3 \times y\); cube root of \(y^3\) is \(y\)The simplification of the second term becomes:\[4x \times 3y \sqrt[3]{3} = 12xy \sqrt[3]{3}\]

Step 2: Check for Common Terms

Both terms, \(4xy \sqrt[3]{3y}\) and \(12xy \sqrt[3]{3}\), seem different due to the presence of \(y\) under the radical in the first term.Neither the coefficients nor the expressions under the radicals can combine directly. Therefore, no further simplification is required, and these terms do not combine into a single term.

Step 3: Write the Final Expression

Since the terms cannot be combined, and the radicals inside the expressions differ, our result will be an expression with simplified individual terms:\[ 4xy \sqrt[3]{3y} + 12xy \sqrt[3]{3}\]

Key Concepts

Radical Expressions

A radical expression includes a root symbol, most commonly a square or cube root. In an expression like \(2 \sqrt[3]{24x^3y^4}\), the cube root symbol \(\sqrt[3]{}\) signifies that you should find the quantity that, when multiplied by itself three times, equals the number or expression inside the root. Radical expressions are frequently seen in algebra and can include variables and coefficients.

These expressions can often be simplified by factoring numbers or expressions inside the root into smaller components. This makes it easier to find multiples or factors that can be freed from the root entirely, leading to a more concise expression.

Understanding how to work with radicals is crucial as they frequently appear in equations across algebra, calculus, and even real-life applications in physics and engineering. Learning to simplify them is a foundational skill in tackling more complex algebraic problems.

Simplifying Radicals

To simplify radicals, start by factorizing the number or expression under the radical into its basic prime components. Let's take the term \( \sqrt[3]{24x^3y^4} \) as an example. You would break it down into compound elements:

• Factorize 24 into \(2^3 \times 3\)
• Recognize \(x^3\) enables the cube root to extract \(x\)
• Factor \(y^4\) as \(y^3 \times y\), with \(y\) extracted from the cube root

By simplifying this radical, we end up with \(4xy \sqrt[3]{3y}\). The goal is to express the radical in a simplified form where as many elements as possible are outside the root. This improves its manageability and opens the door for combining like terms if present.

Simplifying radicals is an essential step before attempting to add or subtract them because only like radicals—those with the same root and radicand—can be directly combined. Thus, simplification can often determine the straightforwardness of solving an algebraic expression.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations such as addition, subtraction, multiplication, and division. In the context of radicals, like in \(2 \sqrt[3]{24 x^{3} y^{4}} + 4 x \sqrt[3]{81 y^{4}}\), each term of the expression can contain radical components, coefficients, and variables.

Working with algebraic expressions involves understanding how to manipulate these terms. One key aspect is identifying and performing operations with like terms. Like terms are terms that have the same variable and exponent component, allowing us to combine them through addition or subtraction.

In situations where the terms involve different radicals that cannot be simplified to become identical, like in our example, each term remains distinct; they cannot be added or subtracted directly. Recognizing this ensures proper handling of such expressions without incorrectly merging non-like terms.

Mastering algebraic expressions with radicals is fundamental for progressing to more advanced math topics, where expression manipulation becomes more sophisticated and strategically vital.