Simplify The Expression: $\[ \left(\sqrt{121} X^3\right)\left(4 \sqrt{25} X^3\right) \\]

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we will focus on simplifying the given expression ${ \left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) }$. We will use various mathematical techniques and properties to simplify the expression and arrive at the final result.

Understanding the Expression

The given expression is a product of two terms, each containing a square root and a variable raised to a power. The first term is $\sqrt{121} x^3$, and the second term is $4 \sqrt{25} x^3$. To simplify the expression, we need to understand the properties of square roots and exponents.

Simplifying Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. In the given expression, we have $\sqrt{121}$ and $\sqrt{25}$. We can simplify these square roots by finding their prime factors.

$\sqrt{121} = \sqrt{11^2} = 11$

$\sqrt{25} = \sqrt{5^2} = 5$

Simplifying Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$. In the given expression, we have $x^3$ in both terms. We can simplify the expression by combining the exponents.

Combining Terms

Now that we have simplified the square roots and exponents, we can combine the terms. We can start by multiplying the coefficients, which are the numbers in front of the variables.

$11 \times 4 = 44$

Simplifying the Expression

Now that we have combined the terms, we can simplify the expression by multiplying the variables.

$\left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) = 44 x^6$

Conclusion

In this article, we simplified the given expression ${ \left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) }$ using various mathematical techniques and properties. We started by simplifying the square roots and exponents, and then combined the terms to arrive at the final result. The simplified expression is $44 x^6$.

Final Answer

The final answer is $\boxed{44 x^6}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the square roots: $\sqrt{121} = 11$ and $\sqrt{25} = 5$
2. Simplify the exponents: $x^3$ remains the same
3. Combine the terms: $11 \times 4 = 44$
4. Simplify the expression: $\left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) = 44 x^6$

Frequently Asked Questions

• What is the simplified expression?
• The simplified expression is $44 x^6$.
• How do you simplify square roots?
• You can simplify square roots by finding their prime factors.
• How do you simplify exponents?
• You can simplify exponents by combining them.

Related Topics

• Simplifying algebraic expressions
• Properties of square roots and exponents
• Combining terms

References

• [1] Khan Academy. (n.d.). Simplifying Algebraic Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/simplifying-algebraic-expressions
• [2] Mathway. (n.d.). Simplifying Algebraic Expressions. Retrieved from https://www.mathway.com/subjects/algebra/simplifying-algebraic-expressions

Keywords

• Simplifying algebraic expressions
• Square roots
• Exponents
• Combining terms
• Algebraic expressions
• Mathematical techniques
• Properties of square roots and exponents
  

Introduction

In our previous article, we simplified the given expression ${ \left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) }$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $44 x^6$.

Q: How do you simplify square roots?

A: You can simplify square roots by finding their prime factors. For example, $\sqrt{121} = \sqrt{11^2} = 11$ and $\sqrt{25} = \sqrt{5^2} = 5$.

Q: How do you simplify exponents?

A: You can simplify exponents by combining them. For example, $x^3$ remains the same.

Q: What is the property of combining terms?

A: When combining terms, you can multiply the coefficients and add the exponents. For example, $\left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) = 44 x^6$.

Q: What are some common mistakes when simplifying algebraic expressions?

A: Some common mistakes include:

• Not simplifying square roots
• Not combining exponents
• Not multiplying coefficients
• Not following the order of operations

Q: How do you check if an expression is simplified?

A: You can check if an expression is simplified by:

• Simplifying square roots
• Combining exponents
• Multiplying coefficients
• Following the order of operations

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Solving equations
• Graphing functions
• Modeling real-world problems
• Optimizing systems

Tips and Tricks

• Always simplify square roots before combining terms.
• Always combine exponents before multiplying coefficients.
• Always follow the order of operations.
• Practice, practice, practice!

Common Misconceptions

• Simplifying algebraic expressions is only for math enthusiasts.
• Simplifying algebraic expressions is only for advanced math students.
• Simplifying algebraic expressions is only for solving equations.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it has many real-world applications. By understanding the properties of square roots and exponents, and by following the order of operations, you can simplify algebraic expressions and arrive at the final result. Remember to always practice, practice, practice!

Final Answer

The final answer is $\boxed{44 x^6}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the square roots: $\sqrt{121} = 11$ and $\sqrt{25} = 5$
2. Simplify the exponents: $x^3$ remains the same
3. Combine the terms: $11 \times 4 = 44$
4. Simplify the expression: $\left(\sqrt{121} x^3\right)\left(4 \sqrt{25} x^3\right) = 44 x^6$

Frequently Asked Questions

• What is the simplified expression?
• The simplified expression is $44 x^6$.
• How do you simplify square roots?
• You can simplify square roots by finding their prime factors.
• How do you simplify exponents?
• You can simplify exponents by combining them.

Related Topics

• Simplifying algebraic expressions
• Properties of square roots and exponents
• Combining terms

References

• [1] Khan Academy. (n.d.). Simplifying Algebraic Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/simplifying-algebraic-expressions
• [2] Mathway. (n.d.). Simplifying Algebraic Expressions. Retrieved from https://www.mathway.com/subjects/algebra/simplifying-algebraic-expressions

Keywords

• Simplifying algebraic expressions
• Square roots
• Exponents
• Combining terms
• Algebraic expressions
• Mathematical techniques
• Properties of square roots and exponents