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

What is the Following Difference? A Comprehensive Analysis

In mathematics, algebraic expressions are used to represent various mathematical operations and relationships. When dealing with algebraic expressions, it's essential to understand the rules of simplification and manipulation. In this article, we will delve into the difference between two algebraic expressions, $11 \sqrt{45} - 4 \sqrt{5}$, and explore the various methods to simplify and evaluate this expression.

The given expression is $11 \sqrt{45} - 4 \sqrt{5}$. To simplify this expression, we need to understand the properties of square roots and how to manipulate them. 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.

To simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$, we need to break it down into smaller components. We can start by simplifying the square roots individually.

• $\sqrt{45}$ can be simplified as $\sqrt{9 \times 5}$, which equals $3 \sqrt{5}$.
• $\sqrt{5}$ remains the same.

Now, we can substitute these simplified values back into the original expression:

$11 \sqrt{45} - 4 \sqrt{5}$

$= 11 \times 3 \sqrt{5} - 4 \sqrt{5}$

$= 33 \sqrt{5} - 4 \sqrt{5}$

Now that we have simplified the individual square roots, we can combine like terms. In this case, we have two terms with the same square root, $\sqrt{5}$. We can combine these terms by adding or subtracting their coefficients.

$33 \sqrt{5} - 4 \sqrt{5}$

$= (33 - 4) \sqrt{5}$

$= 29 \sqrt{5}$

In conclusion, the difference between $11 \sqrt{45} - 4 \sqrt{5}$ is $29 \sqrt{5}$. This result can be obtained by simplifying the individual square roots, combining like terms, and applying the rules of algebraic manipulation.

The correct answer is:

• C. $29 \sqrt{5}$

When dealing with algebraic expressions, it's essential to understand the properties of square roots and how to manipulate them. Here are some additional tips and tricks to help you simplify and evaluate algebraic expressions:

• Simplify individual square roots: Before combining like terms, simplify individual square roots by factoring out perfect squares.
• Combine like terms: Combine terms with the same square root by adding or subtracting their coefficients.
• Apply the rules of algebraic manipulation: Apply the rules of algebraic manipulation, such as the distributive property and the commutative property, to simplify and evaluate algebraic expressions.

By following these tips and tricks, you can simplify and evaluate algebraic expressions with confidence and accuracy.

When dealing with algebraic expressions, it's essential to avoid common mistakes that can lead to incorrect results. Here are some common mistakes to avoid:

• Not simplifying individual square roots: Failing to simplify individual square roots can lead to incorrect results.
• Not combining like terms: Failing to combine like terms can lead to incorrect results.
• Not applying the rules of algebraic manipulation: Failing to apply the rules of algebraic manipulation can lead to incorrect results.

By avoiding these common mistakes, you can ensure that your results are accurate and reliable.

Algebraic expressions have numerous real-world applications in various fields, including science, engineering, economics, and finance. Here are some real-world applications of algebraic expressions:

• Physics and Engineering: Algebraic expressions are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
• Economics: Algebraic expressions are used to model economic systems, predict economic trends, and analyze the behavior of economic variables.
• Finance: Algebraic expressions are used to calculate interest rates, investment returns, and risk management strategies.

By understanding algebraic expressions and how to simplify and evaluate them, you can apply these concepts to real-world problems and make informed decisions.

In conclusion, the difference between $11 \sqrt{45} - 4 \sqrt{5}$ is $29 \sqrt{5}$. This result can be obtained by simplifying the individual square roots, combining like terms, and applying the rules of algebraic manipulation. By following these steps and avoiding common mistakes, you can simplify and evaluate algebraic expressions with confidence and accuracy.
Frequently Asked Questions (FAQs) About Algebraic Expressions

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. Algebraic expressions are used to represent various mathematical relationships and operations.

A: The basic components of an algebraic expression are:

• Variables: Letters or symbols that represent unknown values.
• Constants: Numbers that do not change value.
• Mathematical operations: Addition, subtraction, multiplication, and division.

A: To simplify an algebraic expression, follow these steps:

1. Simplify individual terms: Simplify each term in the expression by combining like terms and applying the rules of algebraic manipulation.
2. Combine like terms: Combine terms with the same variable and coefficient.
3. Apply the rules of algebraic manipulation: Apply the rules of algebraic manipulation, such as the distributive property and the commutative property, to simplify the expression.

A: The difference between $11 \sqrt{45} - 4 \sqrt{5}$ and $29 \sqrt{5}$ is that the former expression is a simplified version of the latter expression. To simplify the expression $11 \sqrt{45} - 4 \sqrt{5}$, we need to break it down into smaller components, simplify the individual square roots, and combine like terms.

A: To evaluate an algebraic expression, follow these steps:

1. Substitute values: Substitute the given values into the expression.
2. Simplify the expression: Simplify the expression by combining like terms and applying the rules of algebraic manipulation.
3. Calculate the result: Calculate the result of the expression.

A: Some common mistakes to avoid when working with algebraic expressions include:

• Not simplifying individual terms: Failing to simplify individual terms can lead to incorrect results.
• Not combining like terms: Failing to combine like terms can lead to incorrect results.
• Not applying the rules of algebraic manipulation: Failing to apply the rules of algebraic manipulation can lead to incorrect results.

A: To apply the rules of algebraic manipulation, follow these steps:

1. Distribute: Distribute the coefficients to the variables.
2. Combine like terms: Combine terms with the same variable and coefficient.
3. Apply the commutative property: Apply the commutative property to rearrange the terms.

A: Algebraic expressions have numerous real-world applications in various fields, including science, engineering, economics, and finance. Some examples of real-world applications of algebraic expressions include:

• Physics and Engineering: Algebraic expressions are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
• Economics: Algebraic expressions are used to model economic systems, predict economic trends, and analyze the behavior of economic variables.
• Finance: Algebraic expressions are used to calculate interest rates, investment returns, and risk management strategies.

By understanding algebraic expressions and how to simplify and evaluate them, you can apply these concepts to real-world problems and make informed decisions.