Select The Correct Answer.What Is This Expression In Simplified Form?\[$(6 \sqrt{2})(-3 \sqrt{5})\$\]A. -90 B. \[$-18 \sqrt{10}\$\] C. \[$-18 \sqrt{7}\$\] D. \[$3 \sqrt{7}\$\]
Understanding the Basics of Radical Expressions
Radical expressions are mathematical expressions that contain a square root or other root. Simplifying radical expressions is an essential skill in mathematics, as it allows us to rewrite complex expressions in a simpler form. In this article, we will focus on simplifying the expression {(6 \sqrt{2})(-3 \sqrt{5})$}$ and selecting the correct answer from the given options.
The Product of Two Radical Expressions
When multiplying two radical expressions, we can simplify the result by multiplying the numbers inside the radicals and then simplifying the resulting expression. In this case, we have the expression {(6 \sqrt{2})(-3 \sqrt{5})$}$. To simplify this expression, we will multiply the numbers inside the radicals and then simplify the resulting expression.
Multiplying the Numbers Inside the Radicals
To multiply the numbers inside the radicals, we will multiply the numbers outside the radicals and then multiply the numbers inside the radicals. In this case, we have:
{(6 \sqrt{2})(-3 \sqrt{5}) = (6)(-3)(\sqrt{2})(\sqrt{5})$}$
Simplifying the Resulting Expression
Now that we have multiplied the numbers inside the radicals, we can simplify the resulting expression. To do this, we will multiply the numbers outside the radicals and then simplify the resulting expression.
{(6)(-3)(\sqrt{2})(\sqrt{5}) = -18(\sqrt{2})(\sqrt{5})$}$
Simplifying the Radical Expression
Now that we have multiplied the numbers outside the radicals, we can simplify the resulting radical expression. To do this, we will multiply the numbers inside the radicals and then simplify the resulting expression.
{-18(\sqrt{2})(\sqrt{5}) = -18\sqrt{10}$}$
Selecting the Correct Answer
Now that we have simplified the expression {(6 \sqrt{2})(-3 \sqrt{5})$}$, we can select the correct answer from the given options. The correct answer is:
{-18 \sqrt{10}$}$
Conclusion
Simplifying radical expressions is an essential skill in mathematics, as it allows us to rewrite complex expressions in a simpler form. In this article, we focused on simplifying the expression {(6 \sqrt{2})(-3 \sqrt{5})$}$ and selecting the correct answer from the given options. By following the steps outlined in this article, you can simplify radical expressions and select the correct answer with confidence.
Final Answer
Q: What is a radical expression?
A: A radical expression is a mathematical expression that contains a square root or other root. Radical expressions are used to represent numbers that are not perfect squares.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to multiply the numbers inside the radicals and then simplify the resulting expression. You can also use the properties of radicals, such as the product rule and the quotient rule, to simplify the expression.
Q: What is the product rule for radicals?
A: The product rule for radicals states that the product of two radical expressions is equal to the product of the numbers inside the radicals. In other words, {\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$}$.
Q: What is the quotient rule for radicals?
A: The quotient rule for radicals states that the quotient of two radical expressions is equal to the quotient of the numbers inside the radicals. In other words, {\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$}$.
Q: How do I simplify an expression with multiple radicals?
A: To simplify an expression with multiple radicals, you need to multiply the numbers inside the radicals and then simplify the resulting expression. You can also use the properties of radicals, such as the product rule and the quotient rule, to simplify the expression.
Q: What is the difference between a rational number and an irrational number?
A: A rational number is a number that can be expressed as the ratio of two integers, such as {\frac{3}{4}$}$. An irrational number is a number that cannot be expressed as the ratio of two integers, such as {\sqrt{2}$}$.
Q: How do I determine if a number is rational or irrational?
A: To determine if a number is rational or irrational, you need to check if it can be expressed as the ratio of two integers. If it can be expressed as the ratio of two integers, it is a rational number. If it cannot be expressed as the ratio of two integers, it is an irrational number.
Q: What is the importance of simplifying radical expressions?
A: Simplifying radical expressions is important because it allows us to rewrite complex expressions in a simpler form. This can make it easier to solve equations and inequalities, and it can also help us to understand the properties of radical expressions.
Q: How do I apply the properties of radicals to simplify expressions?
A: To apply the properties of radicals to simplify expressions, you need to use the product rule and the quotient rule. You can also use the properties of radicals, such as the fact that the product of two radical expressions is equal to the product of the numbers inside the radicals.
Q: What are some common mistakes to avoid when simplifying radical expressions?
A: Some common mistakes to avoid when simplifying radical expressions include:
- Not multiplying the numbers inside the radicals
- Not using the product rule and the quotient rule
- Not simplifying the resulting expression
- Not checking if the number is rational or irrational
Q: How do I check if a number is rational or irrational?
A: To check if a number is rational or irrational, you need to check if it can be expressed as the ratio of two integers. If it can be expressed as the ratio of two integers, it is a rational number. If it cannot be expressed as the ratio of two integers, it is an irrational number.
Q: What are some real-world applications of simplifying radical expressions?
A: Simplifying radical expressions has many real-world applications, including:
- Calculating the area and perimeter of shapes with radical dimensions
- Solving equations and inequalities with radical expressions
- Understanding the properties of radical expressions and their applications in mathematics and science.