Simplify The Expression:$\[ -2 \sqrt{27} - \sqrt{18} + 3 \sqrt{72} \\]

Introduction

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a radical expression in its simplest form, which can be achieved by simplifying the radicand (the number inside the radical sign) and combining like terms. In this article, we will simplify the expression $-2 \sqrt{27} - \sqrt{18} + 3 \sqrt{72}$ using various techniques.

Understanding the Basics of Radical Expressions

Before we dive into simplifying the given expression, let's review the basics of radical expressions. A radical expression is a mathematical expression that contains a square root or a higher root of a number. The square root of a number is denoted by the symbol $\sqrt{}$, and it represents a value that, when multiplied by itself, gives the original number. For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$.

Simplifying the Radicand

To simplify the given expression, we need to simplify the radicand of each term. The radicand is the number inside the radical sign. Let's simplify each term separately.

Simplifying the First Term: $-2 \sqrt{27}$

The first term is $-2 \sqrt{27}$. To simplify this term, we need to find the prime factorization of 27. The prime factorization of 27 is $3^3$. Therefore, we can rewrite the first term as:

$$-2 \sqrt{27} = -2 \sqrt{3^3} = -2 \cdot 3 \sqrt{3} = -6 \sqrt{3}$$

Simplifying the Second Term: $-\sqrt{18}$

The second term is $-\sqrt{18}$. To simplify this term, we need to find the prime factorization of 18. The prime factorization of 18 is $2 \cdot 3^2$. Therefore, we can rewrite the second term as:

$$-\sqrt{18} = -\sqrt{2 \cdot 3^2} = -3 \sqrt{2}$$

Simplifying the Third Term: $3 \sqrt{72}$

The third term is $3 \sqrt{72}$. To simplify this term, we need to find the prime factorization of 72. The prime factorization of 72 is $2^3 \cdot 3^2$. Therefore, we can rewrite the third term as:

$$3 \sqrt{72} = 3 \sqrt{2^3 \cdot 3^2} = 3 \cdot 2 \sqrt{3^2} = 6 \cdot 3 = 18$$

Combining Like Terms

Now that we have simplified each term, we can combine like terms to simplify the expression. The expression can be rewritten as:

$$-6 \sqrt{3} - 3 \sqrt{2} + 18$$

Final Answer

The final answer is $-6 \sqrt{3} - 3 \sqrt{2} + 18$.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it requires a good understanding of the basics of radical expressions. By simplifying the radicand and combining like terms, we can simplify complex radical expressions. In this article, we simplified the expression $-2 \sqrt{27} - \sqrt{18} + 3 \sqrt{72}$ using various techniques. We hope that this article has provided a clear and concise guide to simplifying radical expressions.

Frequently Asked Questions

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to simplify the radicand and combine like terms.

Q: What is the prime factorization of a number?

A: The prime factorization of a number is the expression of the number as a product of prime numbers.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to divide the number by prime numbers until you get a quotient of 1.

Additional Resources

• Simplifying Radical Expressions
• Radical Expressions
• Prime Factorization

References

• Algebra and Trigonometry
• Mathematics for Dummies
• Simplifying Radical Expressions
  

Introduction

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a radical expression in its simplest form, which can be achieved by simplifying the radicand (the number inside the radical sign) and combining like terms. In this article, we will provide a Q&A guide to simplifying radical expressions.

Q&A Guide

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to simplify the radicand and combine like terms.

Q: What is the prime factorization of a number?

A: The prime factorization of a number is the expression of the number as a product of prime numbers.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to divide the number by prime numbers until you get a quotient of 1.

Q: What is the difference between a perfect square and a perfect cube?

A: A perfect square is a number that can be expressed as the square of an integer, while a perfect cube is a number that can be expressed as the cube of an integer.

Q: How do I simplify a radical expression with a perfect square?

A: To simplify a radical expression with a perfect square, you can take the square root of the perfect square and multiply it by the remaining factor.

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

A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction.

Q: How do I simplify a radical expression with a rational root?

A: To simplify a radical expression with a rational root, you can multiply the root by the rational number.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while an exponential expression is a mathematical expression that contains a power of a number.

Q: How do I simplify a radical expression with an exponential expression?

A: To simplify a radical expression with an exponential expression, you can use the property of exponents to simplify the expression.

Examples

Example 1: Simplifying a Radical Expression with a Perfect Square

Simplify the expression $\sqrt{16} + \sqrt{9}$.

Solution: $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$

Example 2: Simplifying a Radical Expression with a Rational Root

Simplify the expression $\sqrt{2} \cdot 3$.

Solution: $\sqrt{2} \cdot 3 = 3\sqrt{2}$

Example 3: Simplifying a Radical Expression with an Exponential Expression

Simplify the expression $\sqrt{2^3} + 2^2$.

Solution: $\sqrt{2^3} + 2^2 = 2 + 4 = 6$

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it requires a good understanding of the basics of radical expressions. By simplifying the radicand and combining like terms, we can simplify complex radical expressions. In this article, we provided a Q&A guide to simplifying radical expressions, including examples and solutions.

Frequently Asked Questions

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to simplify the radicand and combine like terms.

Q: What is the prime factorization of a number?

A: The prime factorization of a number is the expression of the number as a product of prime numbers.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to divide the number by prime numbers until you get a quotient of 1.

Additional Resources

• Simplifying Radical Expressions
• Radical Expressions
• Prime Factorization

References

• Algebra and Trigonometry
• Mathematics for Dummies
• Simplifying Radical Expressions