Simplify The Expression: $\left(2 Y^2\right)\left(4 X Y^3\right) + \left(3 X Y^4\right)(5 Y$\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will focus on simplifying a given expression involving variables and exponents. We will break down the expression into smaller parts, apply the rules of exponents, and combine like terms to arrive at the simplified form.

The Given Expression

The given expression is:

$$\left(2 y^2\right)\left(4 x y^3\right) + \left(3 x y^4\right)(5 y)$$

Step 1: Apply the Distributive Property

To simplify the expression, we will start by applying the distributive property to each term. The distributive property states that for any real numbers a, b, and c:

$$a(b + c) = ab + ac$$

We will apply this property to each term in the given expression.

Distributing the First Term

The first term is $\left(2 y^2\right)\left(4 x y^3\right)$. We can distribute the $2 y^2$ to the $4 x y^3$ as follows:

$$\left(2 y^2\right)\left(4 x y^3\right) = 2 \cdot 4 x y^2 \cdot y^3$$

Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further:

$$2 \cdot 4 x y^2 \cdot y^3 = 8 x y^{2+3} = 8 x y^5$$

Distributing the Second Term

The second term is $\left(3 x y^4\right)(5 y)$. We can distribute the $3 x y^4$ to the $5 y$ as follows:

$$\left(3 x y^4\right)(5 y) = 3 \cdot 5 x y^4 \cdot y$$

Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further:

$$3 \cdot 5 x y^4 \cdot y = 15 x y^{4+1} = 15 x y^5$$

Step 2: Combine Like Terms

Now that we have distributed and simplified each term, we can combine like terms. The expression now looks like this:

$$8 x y^5 + 15 x y^5$$

We can combine the two terms by adding their coefficients:

$$8 x y^5 + 15 x y^5 = (8 + 15) x y^5 = 23 x y^5$$

Conclusion

In this article, we simplified the given expression by applying the distributive property and combining like terms. We started by distributing the terms and simplifying each one using the rules of exponents. Then, we combined the like terms to arrive at the final simplified form. This process demonstrates the importance of simplifying expressions in algebra and how it can help us solve equations and manipulate mathematical statements.

Final Answer

The simplified expression is:

$$23 x y^5$$

Q&A: Simplifying Expressions

In the previous article, we simplified the given expression by applying the distributive property and combining like terms. However, we understand that some readers may still have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses. For example, if we have the expression $(a + b) \cdot c$, we can use the distributive property to expand it as $a \cdot c + b \cdot c$.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the factor outside the parentheses. For example, if we have the expression $(2x + 3y) \cdot 4$, we can apply the distributive property as follows:

$$(2x + 3y) \cdot 4 = 2x \cdot 4 + 3y \cdot 4 = 8x + 12y$$

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. Similarly, $3y$ and $4y$ are like terms because they both have the variable $y$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract their coefficients. For example, if we have the expression $2x^2 + 5x^2$, we can combine the like terms as follows:

$$2x^2 + 5x^2 = (2 + 5)x^2 = 7x^2$$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to distribute the factor outside the parentheses to each term inside the parentheses
• Not combining like terms
• Making errors when multiplying or dividing variables and coefficients
• Not checking the final answer for errors

Q: How can I practice simplifying expressions?

A: There are many ways to practice simplifying expressions, including:

• Working through practice problems in a textbook or online resource
• Creating your own practice problems and solving them
• Using online tools or apps to generate practice problems and check your work
• Asking a teacher or tutor for help and guidance

Conclusion

In this article, we addressed some of the most frequently asked questions about simplifying expressions. We covered topics such as the distributive property, like terms, and common mistakes to avoid. We also provided some tips and resources for practicing simplifying expressions. We hope this article has been helpful in clarifying any confusion and providing a better understanding of simplifying expressions.

Final Tips

• Always read the problem carefully and understand what is being asked
• Use the distributive property to expand expressions
• Combine like terms to simplify expressions
• Check your work for errors
• Practice, practice, practice!

By following these tips and practicing simplifying expressions, you will become more confident and proficient in algebra and be able to tackle more complex problems.