Which Expression Simplifies To $5 \sqrt{3}$?A. $\sqrt{30}$ B. $ 45 \sqrt{45} 45 [/tex] C. $\sqrt{75}$ D. $\sqrt{15}$
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 and apply it to a specific problem. We will examine the given options and determine which expression simplifies to $5 \sqrt{3}$.
Understanding 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 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 Radical Expressions
To simplify a radical expression, we need to find the largest perfect square that divides the number inside the square root. We can then rewrite the expression as the product of the perfect square and the remaining factor.
For example, let's simplify the expression $\sqrt12}$. We can break down 12 into its prime factors \times \sqrt{3}$, which simplifies to 2√3.
Analyzing the Options
Now that we have a basic understanding of simplifying radical expressions, let's examine the given options:
A. $\sqrt{30}$ B. $\sqrt{45}$ C. $\sqrt{75}$ D. $\sqrt{15}$
We need to determine which of these expressions simplifies to $5 \sqrt{3}$.
Option A: $\sqrt{30}$
To simplify $\sqrt{30}$, we need to find the largest perfect square that divides 30. We can break down 30 into its prime factors: 30 = 2 × 3 × 5. Since 2 × 3 is not a perfect square, we cannot simplify the expression further.
Option B: $\sqrt{45}$
To simplify $\sqrt45}$, we need to find the largest perfect square that divides 45. We can break down 45 into its prime factors \times \sqrt{5}$, which simplifies to 3√5.
Option C: $\sqrt{75}$
To simplify $\sqrt75}$, we need to find the largest perfect square that divides 75. We can break down 75 into its prime factors \times \sqrt{3}$, which simplifies to 5√3.
Option D: $\sqrt{15}$
To simplify $\sqrt{15}$, we need to find the largest perfect square that divides 15. We can break down 15 into its prime factors: 15 = 3 × 5. Since 3 × 5 is not a perfect square, we cannot simplify the expression further.
Conclusion
Based on our analysis, we can see that only one of the options simplifies to $5 \sqrt{3}$. The correct answer is:
- C. $\sqrt{75}$
This expression simplifies to $5 \sqrt{3}$, as we can rewrite it as $\sqrt{25} \times \sqrt{3}$, which equals 5√3.
Final Thoughts
Q: What is a radical expression?
A: 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 a value that, when multiplied by itself, gives the original number.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the square root. You can then rewrite the expression as the product of the perfect square and the remaining factor.
Q: What is a perfect square?
A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it can be expressed as 2 × 2.
Q: How do I find the largest perfect square that divides a number?
A: To find the largest perfect square that divides a number, you need to break down the number into its prime factors. You can then identify the perfect square by looking for pairs of identical prime factors.
Q: Can I simplify a radical expression if it has a variable inside the square root?
A: Yes, you can simplify a radical expression if it has a variable inside the square root. However, you need to follow the same steps as before: find the largest perfect square that divides the variable, and then rewrite the expression as the product of the perfect square and the remaining factor.
Q: What if I have a radical expression with multiple terms inside the square root?
A: If you have a radical expression with multiple terms inside the square root, you can simplify it by finding the largest perfect square that divides each term. You can then rewrite the expression as the product of the perfect squares and the remaining factors.
Q: Can I simplify a radical expression if it has a coefficient outside the square root?
A: Yes, you can simplify a radical expression if it has a coefficient outside the square root. However, you need to follow the same steps as before: find the largest perfect square that divides the coefficient, and then rewrite the expression as the product of the perfect square and the remaining factor.
Q: What if I have a radical expression with a negative sign inside the square root?
A: If you have a radical expression with a negative sign inside the square root, you can simplify it by finding the largest perfect square that divides the absolute value of the number. You can then rewrite the expression as the product of the perfect square and the remaining factor, and include the negative sign outside the square root.
Q: Can I simplify a radical expression if it has a fraction inside the square root?
A: Yes, you can simplify a radical expression if it has a fraction inside the square root. However, you need to follow the same steps as before: find the largest perfect square that divides the numerator and the denominator, and then rewrite the expression as the product of the perfect squares and the remaining factors.
Q: What if I have a radical expression with a decimal inside the square root?
A: If you have a radical expression with a decimal inside the square root, you can simplify it by finding the largest perfect square that divides the decimal. However, you may need to use a calculator to find the decimal equivalent of the perfect square.
Conclusion
Simplifying radical expressions is a crucial skill for students to master, and it requires a deep understanding of the underlying mathematics. By following the steps outlined in this article, you can simplify radical expressions and solve complex problems. Remember to always find the largest perfect square that divides the number inside the square root, and then rewrite the expression as the product of the perfect square and the remaining factor.