Problem 86 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[4]{x y^{3}}+\sqrt[4]{x^{5} y} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{1/4} y^{1/4} (y^{1/2} + x) \).

Step 1: Identify and Apply the Fourth Root

First, we identify the two components under each fourth root: \( \sqrt[4]{x y^3} \) and \( \sqrt[4]{x^5 y} \). We can rewrite these components using their exponents: \( \sqrt[4]{x^1 y^3} \) and \( \sqrt[4]{x^5 y^1} \).

Step 2: Simplify Each Component

Using the property of exponents \((\sqrt[n]{a^m} = a^{m/n})\), simplify each component individually:For \( \sqrt[4]{x y^3} \):- Rewrite as \((x y^3)^{1/4}\)- Apply the power rule, so this becomes \(x^{1/4} y^{3/4}\).For \( \sqrt[4]{x^5 y} \):- Rewrite as \((x^5 y)^{1/4}\)- Apply the power rule, so this becomes \(x^{5/4} y^{1/4}\).

Step 3: Combine and Factor Common Terms

Now, combine the simplified expressions:- \(x^{1/4} y^{3/4} + x^{5/4} y^{1/4} \).To factor out common terms, identify the lowest power of each base:- Common base for \(x\) is \(x^{1/4}\)- Common base for \(y\) is \(y^{1/4}\)Factor out \(x^{1/4} y^{1/4}\):\[x^{1/4} y^{1/4} (y^{1/2} + x)\].

Step 4: Final Simplified Expression

The expression is now simplified to factor form:\[x^{1/4} y^{1/4} (y^{1/2} + x)\].

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or fourth roots, like in this exercise. Simplifying these expressions begins by understanding the index or root degree. For example, the fourth root \( \sqrt[4]{a} \) asks which number, multiplied by itself four times, gives \( a \).
To simplify, convert the radical into an expression with exponents. This uses the notation \( a^{1/n} \), where \( n \) represents the root. The problem here uses fourth roots, so \( a^{1/4} \). Identify each term under the radical and express it with fractional exponents. For example, \( \sqrt[4]{x y^3} = (xy^3)^{1/4} \).
This approach displays each term's power clearly, making it easier to simplify or combine expressions. Once in exponent form, further simplification can proceed by adjusting or combining the exponents according to standard algebraic rules.

Exponents and Powers

Exponents tell us how many times a number (the base) is used as a factor. In algebra, managing exponents is crucial. The key is recognizing the rules that guide their manipulation.
When simplifying expressions with exponents and roots, convert roots to fractional exponents. For example, \( \sqrt[n]{a^m} = a^{m/n} \). This transformation allows for the use of standard exponent rules such as:

• \( a^{m} \cdot a^{n} = a^{m+n} \)
• \( (a^{m})^{n} = a^{m \cdot n} \)
• \( \frac{a^{m}}{a^{n}} = a^{m-n} \)

With these rules, simplify each part of a complex expression separately. In our example, \( \sqrt[4]{x y^3} = x^{1/4} y^{3/4} \) and \( \sqrt[4]{x^5 y} = x^{5/4} y^{1/4} \). Applying these rules systematically helps in reaching the simplest form possible for any term.

Factoring Algebraic Expressions

Factoring means breaking down an expression into simpler "factors" that, when multiplied together, return the original expression. It simplifies solving and simplifying complex algebraic expressions.
In our exercise, we first express the terms using their simplest exponents. This setup makes it easier to spot common factors. For instance, identify the smallest power of each variable: for \( x \), the smallest power is \( x^{1/4} \) and for \( y \), it's \( y^{1/4} \).
By factoring out these common terms, the expression \( x^{1/4} y^{1/4} (y^{1/2} + x) \) is achieved. Factoring not only simplifies complex expressions but also reveals similarities between terms that might not be immediately visible, making further manipulations easier.