Select The Correct Answer.What Is This Expression In Simplified Form? $(\sqrt{22})(5 \sqrt{2})$A. $10 \sqrt{11}$ B. $ 12 11 12 \sqrt{11} 12 11 [/tex] C. $20 \sqrt{11}$ D. $24 \sqrt{11}$
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression: $(\sqrt{22})(5 \sqrt{2})$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the reasoning behind each step.
Understanding Radical Expressions
Before we dive into the simplification process, let's take a moment to understand what radical expressions are. A radical expression is a mathematical expression that contains a square root or other root of a number. 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 multiplied by 4 equals 16.
Simplifying the Given Expression
Now that we have a basic understanding of radical expressions, let's focus on simplifying the given expression: $(\sqrt{22})(5 \sqrt{2})$. To simplify this expression, we need to follow the rules of multiplying radical expressions.
Step 1: Multiply the Numbers Inside the Radicals
When multiplying radical expressions, we can multiply the numbers inside the radicals together. In this case, we have:
Step 2: Simplify the Expression Inside the Radicals
Now that we have multiplied the numbers inside the radicals, we can simplify the expression inside the radicals. To do this, we need to look for any common factors between the numbers inside the radicals.
In this case, we can see that both 22 and 2 have a common factor of 2. We can rewrite 22 as 2 x 11, and 2 as 2 x 1. Now we can simplify the expression inside the radicals:
Step 3: Simplify the Radicals
Now that we have simplified the expression inside the radicals, we can simplify the radicals themselves. To do this, we can use the rule that states: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$
In this case, we have:
Step 4: Simplify the Expression
Now that we have simplified the radicals, we can simplify the expression itself. To do this, we can multiply the numbers together:
Step 5: Simplify the Square Root
Finally, we can simplify the square root by taking the square root of the number inside the radical:
Step 6: Multiply the Numbers
Now that we have simplified the square root, we can multiply the numbers together:
Conclusion
In conclusion, simplifying radical expressions requires a step-by-step approach. By following the rules of multiplying radical expressions and simplifying the expression inside the radicals, we can simplify the given expression: $(\sqrt{22})(5 \sqrt{2})$ to $10 \sqrt{11}$. This process requires a clear understanding of radical expressions and the rules for multiplying them.
Answer
The correct answer is:
A. $10 \sqrt{11}$
Discussion
Introduction
In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression: $(\sqrt{22})(5 \sqrt{2})$. We broke down the steps involved in simplifying this expression and provided a clear explanation of the reasoning behind each step. In this article, we will continue to explore the topic of simplifying radical expressions, this time in the form of a 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 other root of a number. A rational expression, on the other hand, is a mathematical expression that contains a fraction. While both types of expressions can be simplified, the process of simplifying radical expressions is distinct from the process of simplifying rational expressions.
Q: How do I simplify a radical expression with multiple terms?
A: To simplify a radical expression with multiple terms, you can follow the same steps as before: multiply the numbers inside the radicals together, simplify the expression inside the radicals, simplify the radicals themselves, and finally simplify the expression. For example, consider the expression: $(\sqrt3})(\sqrt{5})(\sqrt{7})$. To simplify this expression, you would multiply the numbers inside the radicals together)(\sqrt{5})(\sqrt{7}) = \sqrt{3 \cdot 5 \cdot 7} = \sqrt{105}$.
Q: Can I simplify a radical expression with a negative number inside the radical?
A: Yes, you can simplify a radical expression with a negative number inside the radical. To do this, you can use the rule that states: $\sqrt-a} = i \sqrt{a}$, where $i$ is the imaginary unit. For example, consider the expression$. To simplify this expression, you would use the rule: $\sqrt{-16} = i \sqrt{16} = i \cdot 4 = 4i$.
Q: How do I simplify a radical expression with a variable inside the radical?
A: To simplify a radical expression with a variable inside the radical, you can follow the same steps as before: multiply the numbers inside the radicals together, simplify the expression inside the radicals, simplify the radicals themselves, and finally simplify the expression. For example, consider the expression: $\sqrt2x}$$. To simplify this expression, you would multiply the numbers inside the radicals together = \sqrt{2} \cdot \sqrt{x}$.
Q: Can I simplify a radical expression with a fraction inside the radical?
A: Yes, you can simplify a radical expression with a fraction inside the radical. To do this, you can use the rule that states: $\sqrt\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. For example, consider the expression9}}$. To simplify this expression, you would use the rule{9}} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3}$.
Q: How do I simplify a radical expression with multiple radicals?
A: To simplify a radical expression with multiple radicals, you can follow the same steps as before: multiply the numbers inside the radicals together, simplify the expression inside the radicals, simplify the radicals themselves, and finally simplify the expression. For example, consider the expression: $(\sqrt2})(\sqrt{3})(\sqrt{5})$. To simplify this expression, you would multiply the numbers inside the radicals together)(\sqrt{3})(\sqrt{5}) = \sqrt{2 \cdot 3 \cdot 5} = \sqrt{30}$.
Conclusion
In conclusion, simplifying radical expressions requires a clear understanding of the rules for multiplying radical expressions and simplifying the expression inside the radicals. By following these steps, you can simplify a wide range of radical expressions, from simple expressions like $(\sqrt{2})(\sqrt{3})$ to more complex expressions like $(\sqrt{2})(\sqrt{3})(\sqrt{5})$. We hope that this Q&A guide has been helpful in answering your questions about simplifying radical expressions.