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

Understanding the Problem

The given problem involves simplifying an expression that contains square roots. We are required to find the difference between two terms, which are expressed in terms of square roots. The expression is: $11 \sqrt{45} - 4 \sqrt{5}$.

Breaking Down the Expression

To simplify the expression, we need to break it down into smaller parts. We can start by simplifying the square roots individually. The square root of 45 can be simplified as $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$.

Simplifying the Expression

Now that we have simplified the square root of 45, we can substitute this value back into the original expression. The expression becomes: $11 \times 3 \sqrt{5} - 4 \sqrt{5}$.

Combining Like Terms

We can combine the like terms in the expression. The terms $11 \times 3 \sqrt{5}$ and $-4 \sqrt{5}$ can be combined as: $(11 \times 3 - 4) \sqrt{5}$.

Evaluating the Expression

Now we can evaluate the expression by multiplying 11 and 3, and then subtracting 4. The expression becomes: $(33 - 4) \sqrt{5} = 29 \sqrt{5}$.

Comparing the Options

We have simplified the expression to $29 \sqrt{5}$. Now we need to compare this result with the given options. The options are: A. $7 \sqrt{40}$, B. $14 \sqrt{10}$, C. $29 \sqrt{5}$, and D. $95 \sqrt{5}$.

Conclusion

Based on our simplification, we can see that the correct answer is option C. $29 \sqrt{5}$.

Final Answer

The final answer is option C. $29 \sqrt{5}$.

Understanding Square Roots

What are 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.

Simplifying Square Roots

To simplify a square root, we need to find the largest perfect square that divides the number inside the square root. For example, the square root of 45 can be simplified as $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$.

Properties of Square Roots

There are several properties of square roots that we need to know in order to simplify expressions. These properties include:

• The square root of a product is equal to the product of the square roots: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• The square root of a quotient is equal to the quotient of the square roots: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
• The square root of a number is equal to the number raised to the power of 1/2: $\sqrt{a} = a^{\frac{1}{2}}$

Examples of Simplifying Square Roots

Here are some examples of simplifying square roots:

• $\sqrt{16} = 4$
• $\sqrt{25} = 5$
• $\sqrt{36} = 6$
• $\sqrt{49} = 7$
• $\sqrt{64} = 8$
• $\sqrt{81} = 9$
• $\sqrt{100} = 10$

Simplifying Expressions with Square Roots

To simplify an expression with square roots, we need to combine the like terms and then simplify the square roots. For example, the expression $11 \sqrt{45} - 4 \sqrt{5}$ can be simplified as follows:

• Simplify the square root of 45: $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$
• Substitute the simplified square root back into the expression: $11 \times 3 \sqrt{5} - 4 \sqrt{5}$
• Combine the like terms: $(11 \times 3 - 4) \sqrt{5} = 29 \sqrt{5}$

Conclusion

In conclusion, simplifying expressions with square roots requires us to combine the like terms and then simplify the square roots. We need to know the properties of square roots and how to simplify them in order to simplify expressions.

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

A: A square root 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. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. For example, the square root of 45 can be simplified as $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$.

Q: What are some common mistakes to avoid when simplifying expressions with square roots?

A: Some common mistakes to avoid when simplifying expressions with square roots include:

• Not simplifying the square root of a number that can be simplified
• Not combining like terms
• Not using the properties of square roots correctly

Q: How do I use the properties of square roots to simplify expressions?

A: The properties of square roots include:

• The square root of a product is equal to the product of the square roots: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• The square root of a quotient is equal to the quotient of the square roots: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
• The square root of a number is equal to the number raised to the power of 1/2: $\sqrt{a} = a^{\frac{1}{2}}$

Q: Can I simplify an expression with a square root that has a variable inside the square root?

A: Yes, you can simplify an expression with a square root that has a variable inside the square root. However, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root.

Q: How do I simplify an expression with multiple square roots?

A: To simplify an expression with multiple square roots, you need to combine the like terms and then simplify the square roots. For example, the expression $11 \sqrt{45} - 4 \sqrt{5}$ can be simplified as follows:

• Simplify the square root of 45: $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$
• Substitute the simplified square root back into the expression: $11 \times 3 \sqrt{5} - 4 \sqrt{5}$
• Combine the like terms: $(11 \times 3 - 4) \sqrt{5} = 29 \sqrt{5}$

Q: Can I use a calculator to simplify expressions with square roots?

A: Yes, you can use a calculator to simplify expressions with square roots. However, it's always a good idea to check your work by hand to make sure you understand the steps involved in simplifying the expression.

Q: How do I know if an expression with a square root can be simplified?

A: An expression with a square root can be simplified if the number inside the square root can be simplified. For example, the expression $\sqrt{45}$ can be simplified as $\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$.

Q: Can I simplify an expression with a square root that has a negative number inside the square root?

A: No, you cannot simplify an expression with a square root that has a negative number inside the square root. The square root of a negative number is not a real number.

Q: How do I simplify an expression with a square root that has a fraction inside the square root?

A: To simplify an expression with a square root that has a fraction inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{\frac{9}{16}}$ can be simplified as follows:

• Simplify the fraction: $\frac{9}{16}$
• Take the square root of the fraction: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$

Q: Can I simplify an expression with a square root that has a decimal number inside the square root?

A: No, you cannot simplify an expression with a square root that has a decimal number inside the square root. The square root of a decimal number is not a whole number.

Q: How do I simplify an expression with a square root that has a variable with an exponent inside the square root?

A: To simplify an expression with a square root that has a variable with an exponent inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{x^2}$ can be simplified as follows:

• Simplify the exponent: $x^2$
• Take the square root of the exponent: $\sqrt{x^2} = x$

Q: Can I simplify an expression with a square root that has a negative exponent inside the square root?

A: No, you cannot simplify an expression with a square root that has a negative exponent inside the square root. The square root of a negative exponent is not a real number.

Q: How do I simplify an expression with a square root that has a fraction with a negative exponent inside the square root?

A: To simplify an expression with a square root that has a fraction with a negative exponent inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{\frac{9}{16^{-1}}}$ can be simplified as follows:

• Simplify the fraction: $\frac{9}{16^{-1}} = 9 \times 16 = 144$
• Take the square root of the fraction: $\sqrt{144} = 12$

Q: Can I simplify an expression with a square root that has a variable with a negative exponent inside the square root?

A: No, you cannot simplify an expression with a square root that has a variable with a negative exponent inside the square root. The square root of a variable with a negative exponent is not a real number.

Q: How do I simplify an expression with a square root that has a fraction with a variable inside the square root?

A: To simplify an expression with a square root that has a fraction with a variable inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{\frac{x}{y}}$ can be simplified as follows:

• Simplify the fraction: $\frac{x}{y}$
• Take the square root of the fraction: $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$

Q: Can I simplify an expression with a square root that has a variable with a fraction inside the square root?

A: No, you cannot simplify an expression with a square root that has a variable with a fraction inside the square root. The square root of a variable with a fraction is not a real number.

Q: How do I simplify an expression with a square root that has a variable with a square root inside the square root?

A: To simplify an expression with a square root that has a variable with a square root inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{x\sqrt{y}}$ can be simplified as follows:

• Simplify the square root: $\sqrt{y}$
• Take the square root of the square root: $\sqrt{x\sqrt{y}} = \sqrt{x} \times \sqrt{\sqrt{y}} = \sqrt{x} \times y^{\frac{1}{4}}$

Q: Can I simplify an expression with a square root that has a variable with a negative square root inside the square root?

A: No, you cannot simplify an expression with a square root that has a variable with a negative square root inside the square root. The square root of a variable with a negative square root is not a real number.

Q: How do I simplify an expression with a square root that has a variable with a fraction and a square root inside the square root?

A: To simplify an expression with a square root that has a variable with a fraction and a square root inside the square root, you need to follow the same steps as simplifying an expression with a square root that has a constant inside the square root. For example, the expression $\sqrt{\frac{x}{y}\sqrt{z}}$ can be simplified as follows:

• Simplify the fraction: $\frac{x}{y}$
• Simplify the square root: $\sqrt{z}$
• Take the square root of the fraction and the square root: $\sqrt{\frac{x}{y}\sqrt{z}} = \frac{\sqrt{x}}{\sqrt{y}} \times \sqrt{\sqrt{z}} = \frac{\sqrt{x}}{\sqrt{y}} \times z^{\frac{1}{4}}$

Q: Can I simplify an expression with a square root that has a variable with a negative fraction and a square root inside the square root?