Simplify The Expression: $\[ \left(\sqrt{36 X^2}\right)\left(4 \sqrt{9} X^2\right) \\]

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression ${ \left(\sqrt{36 x^2}\right)\left(4 \sqrt{9} x^2\right) }$. We will break down the expression into smaller parts, apply the necessary rules and techniques, and finally simplify the expression to its simplest form.

Understanding the Expression

The given expression is a product of two terms, each containing a square root. The first term is $\sqrt{36 x^2}$, and the second term is $4 \sqrt{9} x^2$. To simplify this expression, we need to apply the rules of exponents and radicals.

Simplifying the First Term

The first term is $\sqrt{36 x^2}$. We can simplify this term by using the rule that $\sqrt{a^2} = a$. In this case, $a = 6x$, so we can rewrite the first term as $6x$.

Simplifying the Second Term

The second term is $4 \sqrt{9} x^2$. We can simplify this term by using the rule that $\sqrt{a^2} = a$. In this case, $a = 3$, so we can rewrite the second term as $4 \cdot 3 \cdot x^2 = 12x^2$.

Multiplying the Terms

Now that we have simplified both terms, we can multiply them together to get the final expression. We have $6x \cdot 12x^2 = 72x^3$.

Conclusion

In this article, we simplified the given expression ${ \left(\sqrt{36 x^2}\right)\left(4 \sqrt{9} x^2\right) }$ by breaking it down into smaller parts, applying the necessary rules and techniques, and finally multiplying the terms together. The final simplified expression is $72x^3$.

Tips and Tricks

• When simplifying expressions, it's essential to apply the rules of exponents and radicals correctly.
• Use the rule that $\sqrt{a^2} = a$ to simplify square roots.
• When multiplying terms, make sure to multiply the coefficients and the variables correctly.

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals correctly.
• Not simplifying square roots correctly.
• Not multiplying terms correctly.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is essential for solving problems involving motion, energy, and momentum. In engineering, simplifying expressions is crucial for designing and analyzing complex systems.

Final Thoughts

Simplifying algebraic expressions is a challenging but essential skill in mathematics. By understanding the rules and techniques involved, we can simplify complex expressions and solve problems with ease. In this article, we simplified the given expression ${ \left(\sqrt{36 x^2}\right)\left(4 \sqrt{9} x^2\right) }$ by breaking it down into smaller parts, applying the necessary rules and techniques, and finally multiplying the terms together. The final simplified expression is $72x^3$.

Additional Resources

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Frequently Asked Questions

• Q: What is the rule for simplifying square roots? A: The rule for simplifying square roots is that $\sqrt{a^2} = a$.
• Q: How do I multiply terms correctly? A: To multiply terms correctly, make sure to multiply the coefficients and the variables correctly.
• Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include not applying the rules of exponents and radicals correctly, not simplifying square roots correctly, and not multiplying terms correctly.
  

Introduction

In our previous article, we simplified the given expression ${ \left(\sqrt{36 x^2}\right)\left(4 \sqrt{9} x^2\right) }$ by breaking it down into smaller parts, applying the necessary rules and techniques, and finally multiplying the terms together. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Q&A Guide

Q: What is the rule for simplifying square roots?

A: The rule for simplifying square roots is that $\sqrt{a^2} = a$. This means that if you have a square root of a perfect square, you can simplify it by taking the square root of the number inside the square root.

Q: How do I simplify expressions with multiple terms?

A: To simplify expressions with multiple terms, you need to apply the rules of exponents and radicals correctly. You can start by simplifying each term separately and then combine the simplified terms.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $3x$, the number 3 is the coefficient and the letter x is the variable.

Q: How do I multiply terms with different variables?

A: To multiply terms with different variables, you need to multiply the coefficients and the variables separately. For example, in the expression $(2x)(3y)$, you would multiply the coefficients 2 and 3 to get 6, and then multiply the variables x and y to get $xy$.

Q: What is the rule for multiplying exponents?

A: The rule for multiplying exponents is that when you multiply two terms with the same base, you add the exponents. For example, in the expression $(x^2)(x^3)$, you would add the exponents 2 and 3 to get $x^5$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you need to move the negative exponent to the other side of the expression. For example, in the expression $\frac{1}{x^2}$, you would rewrite it as $x^{-2}$.

Q: What is the rule for simplifying rational expressions?

A: The rule for simplifying rational expressions is that you can cancel out common factors in the numerator and denominator. For example, in the expression $\frac{6x}{2x}$, you would cancel out the common factor x to get 3.

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals correctly.
• Not simplifying square roots correctly.
• Not multiplying terms correctly.
• Not canceling out common factors in rational expressions.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is essential for solving problems involving motion, energy, and momentum. In engineering, simplifying expressions is crucial for designing and analyzing complex systems.

Final Thoughts

Simplifying algebraic expressions is a challenging but essential skill in mathematics. By understanding the rules and techniques involved, we can simplify complex expressions and solve problems with ease. In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Additional Resources

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Frequently Asked Questions

• Q: What is the rule for simplifying square roots? A: The rule for simplifying square roots is that $\sqrt{a^2} = a$.
• Q: How do I multiply terms correctly? A: To multiply terms correctly, make sure to multiply the coefficients and the variables correctly.
• Q: What are some common mistakes to avoid when simplifying expressions? A: Some common mistakes to avoid when simplifying expressions include not applying the rules of exponents and radicals correctly, not simplifying square roots correctly, and not multiplying terms correctly.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions. By understanding the rules and techniques involved, we can simplify complex expressions and solve problems with ease.