Simplify The Expression: $\[ 9 \sqrt{3} + 2 \sqrt{6} - 7 \sqrt{3} \\]

Feb 26, 2025 by ADMIN 70 views

$Simplify the Expression: $9 \sqrt{3} + 2 \sqrt{6} - 7 \sqrt{3}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand how to combine like terms and simplify radicals. In this article, we will focus on simplifying the given expression: $9 \sqrt{3} + 2 \sqrt{6} - 7 \sqrt{3}$ . We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

Understanding the Expression

The given expression is a combination of three terms: $9 \sqrt{3}$ , $2 \sqrt{6}$ , and $-7 \sqrt{3}$ . To simplify this expression, we need to combine like terms, which means combining terms that have the same variable or radical.

Combining Like Terms

The first step in simplifying the expression is to combine like terms. In this case, we have two terms that contain the variable $\sqrt{3}$ : $9 \sqrt{3}$ and $-7 \sqrt{3}$ . We can combine these terms by adding or subtracting their coefficients.

Combining Coefficients

To combine the coefficients of $9 \sqrt{3}$ and $-7 \sqrt{3}$ , we need to add or subtract them. In this case, we will subtract the coefficient of the second term from the coefficient of the first term:

$9 \sqrt{3} - 7 \sqrt{3} = (9 - 7) \sqrt{3} = 2 \sqrt{3}$

Simplifying the Expression

Now that we have combined the like terms, we can simplify the expression by combining the remaining terms. In this case, we have the term $2 \sqrt{6}$ , which cannot be combined with the other terms.

The simplified expression is:

$2 \sqrt{3} + 2 \sqrt{6}$

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it is crucial to understand how to combine like terms and simplify radicals. In this article, we have focused on simplifying the expression: $9 \sqrt{3} + 2 \sqrt{6} - 7 \sqrt{3}$ . We have broken down the steps involved in simplifying this expression and provided a clear explanation of the process.

Tips and Tricks

When simplifying algebraic expressions, it is essential to combine like terms first.
When combining like terms, add or subtract their coefficients.
Simplify radicals by factoring out perfect squares.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

Physics and Engineering: Simplifying algebraic expressions is essential in physics and engineering, where complex equations need to be solved to understand the behavior of physical systems.
Computer Science: Simplifying algebraic expressions is also essential in computer science, where complex algorithms need to be optimized to improve performance.
Economics: Simplifying algebraic expressions is also essential in economics, where complex models need to be solved to understand the behavior of economic systems.

Common Mistakes

When simplifying algebraic expressions, it is essential to avoid common mistakes, including:

Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
Not simplifying radicals: Failing to simplify radicals can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect conclusions.

Final Thoughts

Additional Resources

For additional resources on simplifying algebraic expressions, including video tutorials and practice problems, please visit the following websites:

Khan Academy: Khan Academy provides video tutorials and practice problems on simplifying algebraic expressions.
Mathway: Mathway provides step-by-step solutions to algebraic expressions, including simplifying radicals.
Wolfram Alpha: Wolfram Alpha provides a comprehensive guide to simplifying algebraic expressions, including video tutorials and practice problems.

Conclusion

Introduction

In our previous article, we simplified the expression: $9 \sqrt{3} + 2 \sqrt{6} - 7 \sqrt{3}$ . We broke down the steps involved in simplifying this expression and provided a clear explanation of the process. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to combine like terms. Like terms are terms that have the same variable or radical.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, if you have the terms $3x$ and $2x$ , you can combine them by adding their coefficients: $3x + 2x = (3 + 2)x = 5x$ .

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

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

Q: How do I simplify radicals?

A: To simplify radicals, factor out perfect squares. For example, if you have the radical $\sqrt{12}$ , you can simplify it by factoring out the perfect square 4: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ .

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as a fraction, such as 3/4 or 22/7. An irrational number is a number that cannot be expressed as a fraction, such as the square root of 2 or the square root of 3.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, factor the numerator and denominator, and then cancel out any common factors. For example, if you have the rational expression $\frac{6x}{2x}$ , you can simplify it by factoring the numerator and denominator: $\frac{6x}{2x} = \frac{3 \cdot 2 \cdot x}{2 \cdot x} = \frac{3 \cdot 2}{2} = 3$ .

Q: What is the difference between a linear and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation $2x + 3 = 5$ is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . For example, if you have the quadratic equation $x^2 + 4x + 4 = 0$ , you can solve it using the quadratic formula: $x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-4 \pm \sqrt{0}}{2} = \frac{-4}{2} = -2$ .

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it is crucial to understand how to combine like terms and simplify radicals. In this article, we have answered some of the most frequently asked questions about simplifying algebraic expressions. By following the tips and tricks provided in this article, you can simplify algebraic expressions with confidence.

Additional Resources

For additional resources on simplifying algebraic expressions, including video tutorials and practice problems, please visit the following websites:

Khan Academy: Khan Academy provides video tutorials and practice problems on simplifying algebraic expressions.
Mathway: Mathway provides step-by-step solutions to algebraic expressions, including simplifying radicals.
Wolfram Alpha: Wolfram Alpha provides a comprehensive guide to simplifying algebraic expressions, including video tutorials and practice problems.

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it is crucial to understand how to combine like terms and simplify radicals. In this article, we have answered some of the most frequently asked questions about simplifying algebraic expressions. By following the tips and tricks provided in this article, you can simplify algebraic expressions with confidence.