Simplify The Expression:$\[ \frac{p^{\frac{1}{2}} Q^2 + \frac{2}{2}}{p^{\frac{1}{4}} Q^{\frac{1}{2}} + \frac{1}{6}} \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a given expression, which involves variables and exponents. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Expression

The given expression is:

$$\frac{p^{\frac{1}{2}} q^2 + \frac{2}{2}}{p^{\frac{1}{4}} q^{\frac{1}{2}} + \frac{1}{6}}$$

At first glance, this expression may seem complex and intimidating. However, with a closer examination, we can identify the individual components and start simplifying.

Step 1: Simplify the Numerator

The numerator of the expression is:

$$p^{\frac{1}{2}} q^2 + \frac{2}{2}$$

We can start by simplifying the second term, which is $\frac{2}{2}$. This is equal to 1, so we can rewrite the numerator as:

$$p^{\frac{1}{2}} q^2 + 1$$

Step 2: Simplify the Denominator

The denominator of the expression is:

$$p^{\frac{1}{4}} q^{\frac{1}{2}} + \frac{1}{6}$$

We can start by simplifying the first term, which is $p^{\frac{1}{4}} q^{\frac{1}{2}}$. This can be rewritten as:

$$\sqrt[4]{p} \sqrt[2]{q}$$

Step 3: Combine Like Terms

Now that we have simplified the numerator and denominator, we can combine like terms. However, in this case, there are no like terms to combine.

Step 4: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression as a whole. We can start by factoring out common terms from the numerator and denominator.

Factoring Out Common Terms

The numerator can be rewritten as:

$$p^{\frac{1}{2}} q^2 + 1$$

The denominator can be rewritten as:

$$\sqrt[4]{p} \sqrt[2]{q} + \frac{1}{6}$$

We can factor out a common term from the numerator and denominator. The common term is $p^{\frac{1}{2}} q^{\frac{1}{2}}$.

Factoring Out $p^{\frac{1}{2}} q^{\frac{1}{2}}$

The numerator can be rewritten as:

$$p^{\frac{1}{2}} q^{\frac{1}{2}} (p q) + 1$$

The denominator can be rewritten as:

$$p^{\frac{1}{2}} q^{\frac{1}{2}} (\sqrt[4]{p} \sqrt[2]{q}) + \frac{1}{6}$$

Canceling Out Common Terms

Now that we have factored out the common term, we can cancel it out. This leaves us with:

$$\frac{p q + 1}{\sqrt[4]{p} \sqrt[2]{q} + \frac{1}{6}}$$

Final Simplification

The expression can be further simplified by rationalizing the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator.

Rationalizing the Denominator

The conjugate of the denominator is:

$$\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6}$$

We can multiply the numerator and denominator by this conjugate:

$$\frac{(p q + 1)(\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})}{(\sqrt[4]{p} \sqrt[2]{q} + \frac{1}{6})(\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})}$$

Simplifying the Expression

Now that we have rationalized the denominator, we can simplify the expression. We can start by expanding the numerator and denominator.

Expanding the Numerator and Denominator

The numerator can be expanded as:

$$(p q + 1)(\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})$$

$$= p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6} + \sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6}$$

The denominator can be expanded as:

$$(\sqrt[4]{p} \sqrt[2]{q} + \frac{1}{6})(\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})$$

$$= (\sqrt[4]{p} \sqrt[2]{q})^2 - (\frac{1}{6})^2$$

$$= p^{\frac{1}{2}} q - \frac{1}{36}$$

Final Simplification

Now that we have expanded the numerator and denominator, we can simplify the expression. We can start by canceling out common terms.

Canceling Out Common Terms

The numerator can be rewritten as:

$$p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6} + \sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6}$$

The denominator can be rewritten as:

$$p^{\frac{1}{2}} q - \frac{1}{36}$$

We can cancel out the common term $p^{\frac{1}{2}} q$ from the numerator and denominator.

Final Simplification

The expression can be simplified as:

$$\frac{p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6} + \sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6}}{p^{\frac{1}{2}} q - \frac{1}{36}}$$

However, this expression can be further simplified by factoring out common terms.

Factoring Out Common Terms

The numerator can be rewritten as:

$$(p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6}) + (\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})$$

The denominator can be rewritten as:

$$p^{\frac{1}{2}} q - \frac{1}{36}$$

We can factor out a common term from the numerator and denominator. The common term is $p^{\frac{1}{2}} q$.

Factoring Out $p^{\frac{1}{2}} q$

The numerator can be rewritten as:

$$(p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6}) + p^{\frac{1}{2}} q (\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})$$

The denominator can be rewritten as:

$$p^{\frac{1}{2}} q (p^{\frac{1}{2}} q - \frac{1}{36})$$

Canceling Out Common Terms

Now that we have factored out the common term, we can cancel it out. This leaves us with:

$$\frac{(p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6}) + p^{\frac{1}{2}} q (\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})}{p^{\frac{1}{2}} q (p^{\frac{1}{2}} q - \frac{1}{36})}$$

Final Simplification

The expression can be simplified as:

$$\frac{(p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6}) + p^{\frac{1}{2}} q (\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})}{p^{\frac{1}{2}} q (p^{\frac{1}{2}} q - \frac{1}{36})}$$

However, this expression can be further simplified by canceling out common terms.

Canceling Out Common Terms

The numerator can be rewritten as:

$$(p q \sqrt[4]{p} \sqrt[2]{q} - \frac{p q}{6}) + p^{\frac{1}{2}} q (\sqrt[4]{p} \sqrt[2]{q} - \frac{1}{6})$$

The denominator can be rewritten as:

$$p^{\frac{1}{2}} q (p^{\frac{1}{2}} q - \frac{1}{36})$$

We can cancel out the common term $p^{\frac{1}{2}} q$ from the numerator and denominator.

Final Simplification

The expression can be simplified as:

\frac{(p q \sqrt[4]{p} \sqrt[<br/> # Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation - Q&A ## Introduction In our previous article, we walked through the process of simplifying a complex expression involving variables and exponents. We broke down the process into manageable steps, making it easier to understand and follow along. In this article, we will address some of the most frequently asked questions related to simplifying expressions. ## Q: What is the first step in simplifying an expression? A: The first step in simplifying an expression is to identify the individual components and simplify each one separately. This may involve factoring out common terms, combining like terms, or simplifying fractions. ## Q: How do I simplify a fraction with variables in the numerator and denominator? A: To simplify a fraction with variables in the numerator and denominator, you can start by factoring out common terms from the numerator and denominator. This may involve canceling out common factors or simplifying fractions. ## Q: What is the difference between simplifying an expression and solving an equation? A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. While simplifying an expression can be a crucial step in solving an equation, they are not the same thing. ## Q: Can I simplify an expression with multiple variables? A: Yes, you can simplify an expression with multiple variables. In fact, simplifying expressions with multiple variables can be a crucial step in solving systems of equations or other types of problems. ## Q: How do I know when an expression is simplified? A: An expression is simplified when it can be reduced to its simplest form, with no further simplification possible. This may involve canceling out common factors, combining like terms, or simplifying fractions. ## Q: Can I use technology to simplify expressions? A: Yes, you can use technology to simplify expressions. Many graphing calculators and computer algebra systems (CAS) can simplify expressions automatically, making it easier to focus on other aspects of the problem. ## Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include: * Not factoring out common terms * Not combining like terms * Not simplifying fractions * Not canceling out common factors * Not checking for errors in the simplification process ## Q: How can I practice simplifying expressions? A: You can practice simplifying expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own, using a calculator or computer algebra system to check your work. ## Q: What are some real-world applications of simplifying expressions? A: Simplifying expressions has many real-world applications, including: * Solving systems of equations * Finding the maximum or minimum value of a function * Optimizing a process or system * Analyzing data and making predictions * Developing mathematical models to describe real-world phenomena ## Conclusion Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify expressions with ease and confidence. Remember to practice regularly and to check your work carefully to avoid common mistakes. ## Additional Resources * Khan Academy: Simplifying Expressions * Mathway: Simplifying Expressions * Wolfram Alpha: Simplifying Expressions ## Final Thoughts Simplifying expressions is a skill that takes practice to develop. By working through examples and exercises, you can become more confident and proficient in simplifying expressions. Remember to always check your work carefully and to use technology to your advantage. With practice and patience, you can master the art of simplifying expressions and tackle even the most complex problems with ease.