Simplify The Following Expression As Much As Possible: − 4 Y 2 × 49 Y 10 -4y^2 \times \sqrt{49y^{10}} − 4 Y 2 × 49 Y 10 ​ Answer: □ \square □ Check Answer

Introduction

In this article, we will delve into the world of algebra and simplify a given expression as much as possible. The expression in question is $-4y^2 \times \sqrt{49y^{10}}$. Our goal is to simplify this expression and provide a clear and concise solution.

Understanding the Expression

Before we begin simplifying the expression, let's break it down and understand its components. The expression consists of two main parts: $-4y^2$ and $\sqrt{49y^{10}}$. The first part is a simple multiplication of $-4$ and $y^2$, while the second part involves the square root of $49y^{10}$.

Simplifying the Square Root

To simplify the square root, we need to find the largest perfect square that divides $49y^{10}$. In this case, the largest perfect square that divides $49y^{10}$ is $49y^8$. We can rewrite $49y^{10}$ as $49y^8 \times y^2$. Therefore, we can simplify the square root as follows:

$$\sqrt{49y^{10}} = \sqrt{49y^8 \times y^2} = \sqrt{49y^8} \times \sqrt{y^2} = 7y^4 \times y = 7y^5$$

Simplifying the Expression

Now that we have simplified the square root, we can substitute this value back into the original expression:

$$-4y^2 \times \sqrt{49y^{10}} = -4y^2 \times 7y^5$$

Using the properties of exponents, we can simplify this expression further:

$$-4y^2 \times 7y^5 = -28y^{2+5} = -28y^7$$

Conclusion

In conclusion, we have successfully simplified the given expression as much as possible. The simplified expression is $-28y^7$. This expression cannot be simplified further, as there are no common factors that can be canceled out.

Final Answer

The final answer is $\boxed{-28y^7}$.

Discussion

This problem involves simplifying an expression using the properties of exponents and square roots. The key to simplifying this expression is to identify the largest perfect square that divides the radicand and then simplify the square root accordingly. Once the square root is simplified, we can substitute this value back into the original expression and simplify further using the properties of exponents.

Related Topics

• Simplifying expressions using the properties of exponents
• Simplifying square roots
• Algebraic manipulation

Example Problems

• Simplify the expression: $3x^2 \times \sqrt{16x^6}$
• Simplify the expression: $2y^3 \times \sqrt{25y^8}$
• Simplify the expression: $5z^4 \times \sqrt{81z^{12}}$

Practice Problems

• Simplify the expression: $-2x^2 \times \sqrt{36x^8}$
• Simplify the expression: $3y^2 \times \sqrt{49y^{10}}$
• Simplify the expression: $4z^3 \times \sqrt{64z^{12}}$

Glossary

• Radicand: The expression under the square root symbol.
• Perfect square: A number that can be expressed as the square of an integer.
• Exponent: A small number that indicates the power to which a base number is raised.
• Simplify: To reduce an expression to its simplest form by canceling out common factors.
  Simplifying the Given Expression: A Q&A Guide ===========================================================

Introduction

In our previous article, we simplified the expression $-4y^2 \times \sqrt{49y^{10}}$ to $-28y^7$. However, we understand that simplifying expressions can be a challenging task, and many of you may have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying expressions.

Q&A

Q: What is the largest perfect square that divides $49y^{10}$?

A: The largest perfect square that divides $49y^{10}$ is $49y^8$. This is because $49y^8$ is the largest number that can be expressed as the square of an integer and still divide $49y^{10}$.

Q: How do I simplify the square root of $49y^{10}$?

A: To simplify the square root of $49y^{10}$, we need to find the largest perfect square that divides $49y^{10}$. In this case, the largest perfect square that divides $49y^{10}$ is $49y^8$. We can rewrite $49y^{10}$ as $49y^8 \times y^2$. Therefore, we can simplify the square root as follows:

$$\sqrt{49y^{10}} = \sqrt{49y^8 \times y^2} = \sqrt{49y^8} \times \sqrt{y^2} = 7y^4 \times y = 7y^5$$

Q: How do I simplify the expression $-4y^2 \times 7y^5$?

A: To simplify the expression $-4y^2 \times 7y^5$, we can use the properties of exponents. Specifically, we can add the exponents of the same base:

$$-4y^2 \times 7y^5 = -28y^{2+5} = -28y^7$$

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

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

• Not identifying the largest perfect square that divides the radicand
• Not using the properties of exponents correctly
• Not canceling out common factors

Q: How can I practice simplifying expressions?

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

• Working through example problems
• Practicing with online resources, such as math websites and apps
• Asking a teacher or tutor for help

Related Topics

• Simplifying expressions using the properties of exponents
• Simplifying square roots
• Algebraic manipulation

Example Problems

• Simplify the expression: $3x^2 \times \sqrt{16x^6}$
• Simplify the expression: $2y^3 \times \sqrt{25y^8}$
• Simplify the expression: $5z^4 \times \sqrt{81z^{12}}$

Practice Problems

• Simplify the expression: $-2x^2 \times \sqrt{36x^8}$
• Simplify the expression: $3y^2 \times \sqrt{49y^{10}}$
• Simplify the expression: $4z^3 \times \sqrt{64z^{12}}$

Glossary

• Radicand: The expression under the square root symbol.
• Perfect square: A number that can be expressed as the square of an integer.
• Exponent: A small number that indicates the power to which a base number is raised.
• Simplify: To reduce an expression to its simplest form by canceling out common factors.

Conclusion

Simplifying expressions can be a challenging task, but with practice and patience, you can become proficient in simplifying expressions. Remember to identify the largest perfect square that divides the radicand, use the properties of exponents correctly, and cancel out common factors. With these tips and practice problems, you will be well on your way to becoming a master of simplifying expressions.