Select The Correct Answer.What Is This Expression In Simplified Form?$5 \sqrt{63} - 6 \sqrt{28}$A. $21 \sqrt{7}$B. \$3 \sqrt{7}$[/tex\]C. $11 \sqrt{7}$D. $27 \sqrt{7}$
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: $5 \sqrt{63} - 6 \sqrt{28}$. 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 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.
Breaking Down the Given Expression
The given expression is: $5 \sqrt{63} - 6 \sqrt{28}$. To simplify this expression, we need to break it down into its individual components. Let's start by simplifying the square roots of 63 and 28.
Simplifying the Square Root of 63
The square root of 63 can be simplified by finding the largest perfect square that divides 63. In this case, the largest perfect square that divides 63 is 9, because 9 multiplied by 7 equals 63. Therefore, we can rewrite the square root of 63 as: $\sqrt{63} = \sqrt{9 \times 7} = 3 \sqrt{7}$.
Simplifying the Square Root of 28
The square root of 28 can be simplified by finding the largest perfect square that divides 28. In this case, the largest perfect square that divides 28 is 4, because 4 multiplied by 7 equals 28. Therefore, we can rewrite the square root of 28 as: $\sqrt{28} = \sqrt{4 \times 7} = 2 \sqrt{7}$.
Substituting the Simplified Square Roots
Now that we have simplified the square roots of 63 and 28, we can substitute these simplified forms back into the original expression. The expression becomes: $5 \sqrt{63} - 6 \sqrt{28} = 5(3 \sqrt{7}) - 6(2 \sqrt{7})$.
Distributing the Coefficients
To simplify the expression further, we need to distribute the coefficients to the terms inside the parentheses. The expression becomes: $15 \sqrt{7} - 12 \sqrt{7}$.
Combining Like Terms
Now that we have distributed the coefficients, we can combine like terms. The expression becomes: $(15 - 12) \sqrt{7} = 3 \sqrt{7}$.
Conclusion
In conclusion, the simplified form of the given expression $5 \sqrt63} - 6 \sqrt{28}$ is $3 \sqrt{7}$. This expression can be further simplified by factoring out the common term $\sqrt{7}$, resulting in the final answer$.
Answer
The correct answer is: B. $3 \sqrt{7}$
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 break it down into its individual components and simplify the square roots. This involves finding the largest perfect square that divides the number inside the square root and rewriting it as a product of the square root of the perfect square and the remaining number.
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 x 2, and 9 is a perfect square because it can be expressed as 3 x 3.
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 look for the largest number that can be expressed as the product of an integer with itself. For example, to find the largest perfect square that divides 63, you would look for the largest number that can be expressed as the product of an integer with itself, which is 9 (3 x 3).
Q: Can I simplify a radical expression with a negative number inside the square root?
A: Yes, you can simplify a radical expression with a negative number inside the square root. However, you need to remember that the square root of a negative number is an imaginary number, which is a complex number that cannot be expressed as a real number.
Q: How do I simplify a radical expression with a variable inside the square root?
A: To simplify a radical expression with a variable inside the square root, you need to follow the same steps as simplifying a radical expression with a number inside the square root. However, you need to be careful when simplifying the expression, as the variable may have different values.
Q: Can I simplify a radical expression with multiple terms inside the square root?
A: Yes, you can simplify a radical expression with multiple terms inside the square root. However, you need to follow the same steps as simplifying a radical expression with a single term inside the square root, and you need to be careful when combining like terms.
Q: How do I know if a radical expression is already simplified?
A: To determine if a radical expression is already simplified, you need to check if the expression contains any perfect squares that can be factored out. If the expression contains any perfect squares that can be factored out, it is not already simplified.
Q: Can I simplify a radical expression with a fraction inside the square root?
A: Yes, you can simplify a radical expression with a fraction inside the square root. However, you need to follow the same steps as simplifying a radical expression with a number inside the square root, and you need to be careful when simplifying the fraction.
Q: How do I simplify a radical expression with a decimal number inside the square root?
A: To simplify a radical expression with a decimal number inside the square root, you need to follow the same steps as simplifying a radical expression with a number inside the square root. However, you need to be careful when simplifying the decimal number, as it may have different values.
Conclusion
Simplifying radical expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By following the steps outlined in this article, you can simplify radical expressions with ease and accuracy. Remember to be careful when simplifying expressions with variables, fractions, and decimal numbers, as they may have different values.