What Is The Following Difference?$11 \sqrt{45} - 4 \sqrt{5}$A. $7 \sqrt{40}$ B. $14 \sqrt{10}$ C. $29 \sqrt{5}$ D. $95 \sqrt{5}$

What is the Following Difference? A Comprehensive Analysis

The given expression is $11 \sqrt{45} - 4 \sqrt{5}$, and we are required to simplify it and find the difference. To do this, we need to understand the properties of square roots and how to simplify expressions involving them.

Understanding Square Roots

A 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. The square root of a number is denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ represents the square root of 16.

Simplifying Expressions Involving Square Roots

To simplify an expression involving square roots, we need to find the square root of each number and then combine the results. For example, $\sqrt{16} + \sqrt{9}$ can be simplified as $4 + 3 = 7$.

Simplifying the Given Expression

Now, let's simplify the given expression $11 \sqrt{45} - 4 \sqrt{5}$. To do this, we need to find the square root of 45 and then simplify the expression.

Finding the Square Root of 45

The square root of 45 can be found by breaking it down into its prime factors. 45 can be written as $9 \times 5$, where 9 is a perfect square. Therefore, $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$.

Simplifying the Expression

Now that we have found the square root of 45, we can simplify the given expression. $11 \sqrt{45} - 4 \sqrt{5}$ can be written as $11 \times 3 \sqrt{5} - 4 \sqrt{5} = 33 \sqrt{5} - 4 \sqrt{5} = 29 \sqrt{5}$.

Conclusion

Therefore, the simplified expression is $29 \sqrt{5}$. This means that the correct answer is C. $29 \sqrt{5}$.

Comparison with Other Options

Let's compare our answer with the other options:

• A. $7 \sqrt{40}$: This option is incorrect because the square root of 40 is not equal to 7.
• B. $14 \sqrt{10}$: This option is incorrect because the square root of 10 is not equal to 14.
• D. $95 \sqrt{5}$: This option is incorrect because the simplified expression is $29 \sqrt{5}$, not $95 \sqrt{5}$.

Final Answer

The final answer is C. $29 \sqrt{5}$.
Frequently Asked Questions (FAQs) About Simplifying Expressions Involving Square Roots

In the previous article, we discussed how to simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$. However, we understand that you may have more questions about simplifying expressions involving square roots. In this article, we will address some of the most frequently asked questions (FAQs) about this topic.

Q: What is the difference between a square root and a radical?

A: A square root and a radical are often used interchangeably, but technically, a radical is a more general term that refers to any root of a number, not just a square root. For example, $\sqrt[3]{x}$ is a radical, but not a square root.

Q: How do I simplify an expression involving a square root?

A: To simplify an expression involving a square root, you need to find the square root of each number and then combine the results. For example, $\sqrt{16} + \sqrt{9}$ can be simplified as $4 + 3 = 7$.

Q: What is the rule for simplifying expressions involving square roots?

A: The rule for simplifying expressions involving square roots is to find the square root of each number and then combine the results. If the numbers have a common factor, you can simplify the expression by factoring out the common factor.

Q: How do I simplify an expression involving a square root of a product?

A: To simplify an expression involving a square root of a product, you need to find the square root of each number and then multiply the results. For example, $\sqrt{9 \times 5}$ can be simplified as $3 \sqrt{5}$.

Q: What is the difference between a perfect square and an imperfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. An imperfect square is a number that cannot be expressed as the square of an integer.

Q: How do I simplify an expression involving a square root of a perfect square?

A: To simplify an expression involving a square root of a perfect square, you can simply take the square root of the perfect square. For example, $\sqrt{16}$ can be simplified as $4$.

Q: What is the rule for simplifying expressions involving square roots of fractions?

A: The rule for simplifying expressions involving square roots of fractions is to simplify the fraction first and then find the square root of the result. For example, $\sqrt{\frac{9}{16}}$ can be simplified as $\frac{3}{4}$.

Q: How do I simplify an expression involving a square root of a negative number?

A: To simplify an expression involving a square root of a negative number, you need to use the imaginary unit $i$, which is defined as the square root of $-1$. For example, $\sqrt{-16}$ can be simplified as $4i$.

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. For example, $\frac{3}{4}$ is a rational number. An irrational number is a number that cannot be expressed as the ratio of two integers.

Q: How do I simplify an expression involving a square root of an irrational number?

A: To simplify an expression involving a square root of an irrational number, you need to use the properties of square roots to simplify the expression. For example, $\sqrt{2}$ is an irrational number, and it cannot be simplified further.

Conclusion

In this article, we have addressed some of the most frequently asked questions (FAQs) about simplifying expressions involving square roots. We hope that this article has provided you with a better understanding of this topic and has helped you to simplify expressions involving square roots with confidence.