Select The Correct Answer.What Is This Expression In Simplified Form? $5 \sqrt{2} \cdot 9 \sqrt{6}$A. $45 \sqrt{2}$ B. $ 45 3 45 \sqrt{3} 45 3 [/tex] C. $90 \sqrt{3}$ D. 90
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression: $5 \sqrt{2} \cdot 9 \sqrt{6}$. 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, it's essential to understand the basics of radical expressions. A radical expression is a mathematical expression that contains a square root or a higher-order root. 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: $5 \sqrt{2} \cdot 9 \sqrt{6}$. To simplify this expression, we need to follow the rules of multiplying radical expressions.
Rule 1: Multiply the Coefficients
The first step in simplifying the given expression is to multiply the coefficients. In this case, the coefficients are the numbers outside the square roots. We multiply 5 and 9 to get 45.
Rule 2: Multiply the Radicals
The next step is to multiply the radicals. When multiplying radical expressions, we multiply the numbers inside the square roots. In this case, we multiply $\sqrt{2}$ and $\sqrt{6}$. To do this, we need to find the product of the numbers inside the square roots.
Finding the Product of the Radicals
To find the product of the radicals, we need to multiply the numbers inside the square roots. In this case, we multiply 2 and 6 to get 12. However, we also need to consider the square root of the product. The square root of 12 can be simplified as $\sqrt{4 \cdot 3}$, which equals $2 \sqrt{3}$.
Combining the Results
Now that we have multiplied the coefficients and the radicals, we can combine the results. We multiply the coefficients (45) and the simplified radical expression ($2 \sqrt{3}$) to get the final result.
The Final Result
The final result of simplifying the given expression is: $45 \cdot 2 \sqrt{3}$, which equals $90 \sqrt{3}$.
Conclusion
In this article, we have explored the process of simplifying radical expressions, with a focus on the given expression: $5 \sqrt2} \cdot 9 \sqrt{6}$. We have broken down the steps involved in simplifying this expression and provided a clear explanation of the reasoning behind each step. By following the rules of multiplying radical expressions, we have arrived at the final result$.
Answer
The correct answer is: C. $90 \sqrt{3}$
Discussion
This expression can be simplified by multiplying the coefficients and the radicals. The coefficients are 5 and 9, which multiply to 45. The radicals are $\sqrt{2}$ and $\sqrt{6}$, which multiply to $\sqrt{12}$. The square root of 12 can be simplified as $\sqrt{4 \cdot 3}$, which equals $2 \sqrt{3}$. Therefore, the final result is $45 \cdot 2 \sqrt{3}$, which equals $90 \sqrt{3}$.
Additional Examples
To further illustrate the process of simplifying radical expressions, let's consider a few additional examples:
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3 \sqrt{5} \cdot 2 \sqrt{5}$: This expression can be simplified by multiplying the coefficients (3 and 2) and the radicals ($\sqrt{5}$ and $\sqrt{5}$). The result is $6 \sqrt{5}$.
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4 \sqrt{2} \cdot 3 \sqrt{2}$: This expression can be simplified by multiplying the coefficients (4 and 3) and the radicals ($\sqrt{2}$ and $\sqrt{2}$). The result is $12 \sqrt{2}$.
Conclusion
Introduction
In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression: $5 \sqrt{2} \cdot 9 \sqrt{6}$. 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 provide a Q&A guide on simplifying radical expressions.
Q: What is the first step in simplifying a radical expression?
A: The first step in simplifying a radical expression is to identify the coefficients and the radicals. The coefficients are the numbers outside the square roots, while the radicals are the numbers inside the square roots.
Q: How do I multiply the coefficients?
A: To multiply the coefficients, simply multiply the numbers outside the square roots. For example, if you have the expression $3 \sqrt{5} \cdot 2 \sqrt{5}$, you would multiply the coefficients (3 and 2) to get 6.
Q: How do I multiply the radicals?
A: To multiply the radicals, you need to multiply the numbers inside the square roots. For example, if you have the expression $\sqrt{2} \cdot \sqrt{6}$, you would multiply the numbers inside the square roots (2 and 6) to get 12. However, you also need to consider the square root of the product, which is $\sqrt{12}$.
Q: How do I simplify the square root of a product?
A: To simplify the square root of a product, you need to find the largest perfect square that divides the product. For example, if you have the expression $\sqrt{12}$, you can simplify it as $\sqrt{4 \cdot 3}$, which equals $2 \sqrt{3}$.
Q: What is the final step in simplifying a radical expression?
A: The final step in simplifying a radical expression is to combine the results of multiplying the coefficients and the radicals. For example, if you have the expression $5 \sqrt{2} \cdot 9 \sqrt{6}$, you would multiply the coefficients (5 and 9) to get 45, and then multiply the radicals ($\sqrt{2}$ and $\sqrt{6}$) to get $\sqrt{12}$, which simplifies to $2 \sqrt{3}$. The final result is $45 \cdot 2 \sqrt{3}$, which equals $90 \sqrt{3}$.
Q: Can I simplify a radical expression with a negative coefficient?
A: Yes, you can simplify a radical expression with a negative coefficient. To do this, you need to multiply the coefficient by the radical. For example, if you have the expression $-3 \sqrt{5}$, you can simplify it as $-3 \cdot \sqrt{5}$.
Q: Can I simplify a radical expression with a fraction as a coefficient?
A: Yes, you can simplify a radical expression with a fraction as a coefficient. To do this, you need to multiply the fraction by the radical. For example, if you have the expression $\frac{1}{2} \sqrt{3}$, you can simplify it as $\frac{1}{2} \cdot \sqrt{3}$.
Q: Can I simplify a radical expression with a variable as a coefficient?
A: Yes, you can simplify a radical expression with a variable as a coefficient. To do this, you need to multiply the variable by the radical. For example, if you have the expression $x \sqrt{2}$, you can simplify it as $x \cdot \sqrt{2}$.
Conclusion
In conclusion, simplifying radical expressions is a crucial skill for students to master. By following the rules of multiplying radical expressions, we can simplify complex expressions and arrive at the final result. In this article, we have provided a Q&A guide on simplifying radical expressions, covering topics such as multiplying coefficients, multiplying radicals, and simplifying the square root of a product. We hope that this guide has been helpful in providing a clear understanding of the process of simplifying radical expressions.