Problem 59 - Algebra A Combined Function

Question

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{32 x^{12} y^{5}} $$

Step-by-Step Solution

Answer

The simplified form is \( 2 x^3 y \cdot \sqrt[4]{2y} \).

Step 1: Express the Radicand as a Product of Powers

To simplify \( \sqrt[4]{32 x^{12} y^{5}} \), first express the number and variables under the radical as powers, if possible. We can write:\[ 32 = 2^5 \]Thus,\[ \sqrt[4]{32 x^{12} y^{5}} = \sqrt[4]{2^5 x^{12} y^{5}} \]

Step 2: Split the Radicand into Separate Radicals

Split the fourth root into a product of separate radicals for numbers and each variable factor:\[ \sqrt[4]{2^5 x^{12} y^{5}} = \sqrt[4]{2^5} \cdot \sqrt[4]{x^{12}} \cdot \sqrt[4]{y^{5}} \]

Step 3: Simplify Each Radical

Simplify each radical by taking the fourth root of the powers:- \( \sqrt[4]{2^5} \) involves splitting into \( 2^4 \times 2^1 \), hence \( \sqrt[4]{2^4} = 2 \).- \( \sqrt[4]{2^5} = 2 \cdot \sqrt[4]{2} \).- \( \sqrt[4]{x^{12}} = x^{12/4} = x^3 \).- \( \sqrt[4]{y^{5}} = y^{5/4} = y^{1.25} = y \cdot y^{1/4} \) (leave the fractional exponent as is for partial simplification).Thus,\[ 2 \cdot \sqrt[4]{2} \cdot x^3 \cdot y \cdot \sqrt[4]{y} \]

Step 4: Combine the Simplified Components

Combine all simplified parts into one expression:\[ 2x^3y \cdot \sqrt[4]{2} \cdot \sqrt[4]{y} \]And as the fourth roots can be further combined:\[ = 2 x^3 y \cdot \sqrt[4]{2y} \]

Step 5: Final Simplification Check

Verify all possible simplifications have been made: The expression \( 2 x^3 y \cdot \sqrt[4]{2y} \) is fully simplified because neither \( 2y \) nor its fourth root can be simplified further with perfect fourth powers.

Key Concepts

Simplifying Radicals

Simplifying radicals means reducing them to their simplest form. This process can make complex expressions easier to understand and work with. To simplify a radical expression like \( \sqrt[4]{32 x^{12} y^{5}} \), we follow these basic steps:

• Express numbers and variables as powers, if possible, under the radical sign.
• Separate the radical into smaller, manageable pieces.
• Simplify each piece by taking the artithmetical or algebraic root.

Remember that the goal is to reduce the expression so that there are no more perfect powers inside the radical, making it cleaner and simpler.

Fourth Roots

The fourth root of a number or expression is another value that, when raised to the power of four, equals the original number. For example, the fourth root of \( 16 \) is \( 2 \) since \( 2^4 = 16 \). When simplifying an expression like \( \sqrt[4]{32} \), you need to identify parts that are perfect fourth powers:

• Split numbers or variables into components that can easily yield perfect fourth powers.
• If a component isn't a perfect fourth power, simplify as much as possible.

Fourth roots are extended versions of square and cube roots, following similar principles of simplification.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They can represent quantities and are a key part of simplifying radicals. When working with an expression like \( \sqrt[4]{32 x^{12} y^{5}} \), understanding how to manipulate and simplify each component is crucial:

• Recognize parts of the expression that can be separated and simplified individually.
• Use laws of exponents to break down and simplify variable components.

Overall, simplifying algebraic expressions within radicals requires careful attention to each part of the expression.

Power of a Variable

In algebra, the power of a variable represents how many times the variable is used as a factor. Simplifying expressions involves manipulating these powers. For instance, in \( x^{12} \):

• Identify how the power compares to the root being applied, here a fourth root.
• Divide the exponent by the root to simplify, such as \( x^{12/4} = x^3 \).
• If division leaves a remainder, simplify to the fullest extent possible.

These steps help in reducing expressions to their simplest form, making calculations manageable and results more intuitive.