Problem 29 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt[3]{54 x y^{3}}-5 \sqrt[3]{2 x y^{3}}+y \sqrt[3]{128 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2y \sqrt[3]{2x} \).

Step 1: Break the Radicals into Prime Factors

First, let's break each term into its prime factors to simplify them. For the first term:\[ \sqrt[3]{54 x y^3} = \sqrt[3]{2 \times 3^3 \times x \times y^3} \]For the second term:\[ 5 \sqrt[3]{2 x y^3} = 5 \times \sqrt[3]{2 \times x \times y^3} \]For the third term:\[ y \sqrt[3]{128 x} = y \times \sqrt[3]{2^7 \times x} \]

Step 2: Simplify Each Radical

Now we will simplify each expression by taking out cubes when possible.First term:\[ \sqrt[3]{54 x y^3} = \sqrt[3]{(3^3) \times 2 \times x \times y^3} = 3y \sqrt[3]{2x} \]Second term:\[ 5 \sqrt[3]{2 x y^3} = 5y \sqrt[3]{2x} \]Third term:\[ y \sqrt[3]{128 x} = y \sqrt[3]{2^7 \times x} = y \times (2^2) \times \sqrt[3]{2x} = 4y \sqrt[3]{2x} \]

Step 3: Combine Like Terms

Now that all terms have \( y \sqrt[3]{2x} \), we can combine them:\[ 3y \sqrt[3]{2x} - 5y \sqrt[3]{2x} + 4y \sqrt[3]{2x} \]Combine the coefficients:\[ (3y - 5y + 4y) \sqrt[3]{2x} = 2y \sqrt[3]{2x} \]

Step 4: Conclusion

The expression simplifies to:\[ 2y \sqrt[3]{2x} \]

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers, which are numbers only divisible by 1 and themselves. Understanding prime factorization is essential when dealing with radicals because it allows for simplifying expressions more easily. For example, consider the number 54. We break it down as follows:

• 54 is divisible by 2 (a prime number), which gives 27
• 27 is divisible by 3, resulting in 9
• 9 is also divisible by 3, and we finally reach 3

Hence, 54 can be expressed as \(2 \times 3^3\).

Performing prime factorization on each component of a radical ensures that we can easily identify and extract perfect cubes, squares, or similar factors. This skill lays the groundwork for further simplification and manipulation of radical expressions.

Simplifying Radicals

Simplifying radicals involves reducing a radical to its simplest form. This is achieved by identifying and extracting perfect square roots (or higher roots like cube roots) from under the radical sign. By using prime factorization, you can determine which parts of the expression can be simplified.

For instance, take \( \sqrt[3]{54xy^3} \). After factorizing, you find \(54 = 2 \times 3^3\), and because \(y^3\) is a perfect cube, you can simplify \( \sqrt[3]{54xy^3} = 3y \sqrt[3]{2x} \). This extraction process applies similarly to the other terms in the example, allowing us to make the radicals uniform and easier to manage.

Remember, simplifying radicals is a crucial step in ensuring the expressions are as straightforward as possible before combining or performing additional operations.

Cubed Roots

Cubed roots specifically refer to finding a number that, when multiplied by itself three times, equals the original number. For example, the cubed root of 8 is 2, because \(2 \times 2 \times 2 = 8\). In algebra, dealing with cubed roots often requires simplifying expressions via factorization and extraction of cubes.

In the problem provided, terms involve cubed roots such as \( \sqrt[3]{54xy^3} \). After factoring, you simplify using the perfect cubes present in the expression. This process reduces expressions to forms that can be manipulated easily, as seen with terms like \( 3y \sqrt[3]{2x} \). Understanding cubed roots and their properties is vital to mastering radical operations and simplifications.

Combining Like Terms

Combining like terms is an important algebraic concept, particularly when simplifying expressions. Like terms are terms that have identical variable parts and therefore can be added or subtracted together. This concept becomes important in dealing with radicals, especially after simplification.

In our example, the expression simplifies so each term includes \(y \sqrt[3]{2x}\). Thus, you have:

• \(3y \sqrt[3]{2x}\)
• - \(5y \sqrt[3]{2x}\)
• \(+ 4y \sqrt[3]{2x}\)

The coefficients (3, -5, and 4) are combined to give \((3 - 5 + 4)y \sqrt[3]{2x}\), simplifying to \(2y \sqrt[3]{2x}\).

Combining like terms simplifies calculations and reduces expression complexity, making your algebraic work clearer and more efficient.