Evaluate { \sqrt{25^2 - 24^2}$}$.

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Introduction

In mathematics, the expression $\sqrt{25^2 - 24^2}$ is a classic example of a difference of squares, which can be simplified using the formula $(a^2 - b^2) = (a + b)(a - b)$. This formula is a fundamental concept in algebra and is used extensively in various mathematical operations. In this article, we will evaluate the expression $\sqrt{25^2 - 24^2}$ using the difference of squares formula and explore its significance in mathematics.

Understanding the Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that states:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be applied to any two numbers $a$ and $b$, and it is used to simplify expressions that involve the difference of squares. In the expression $\sqrt{25^2 - 24^2}$, we can see that it is a difference of squares, where $a = 25$ and $b = 24$.

Evaluating the Expression

To evaluate the expression $\sqrt{25^2 - 24^2}$, we can use the difference of squares formula:

$$\sqrt{25^2 - 24^2} = \sqrt{(25 + 24)(25 - 24)}$$

Using the formula, we can simplify the expression as follows:

$$\sqrt{(25 + 24)(25 - 24)} = \sqrt{(49)(1)}$$

Now, we can simplify the expression further by evaluating the square root:

$$\sqrt{(49)(1)} = \sqrt{49} = 7$$

Therefore, the value of the expression $\sqrt{25^2 - 24^2}$ is 7.

Significance of the Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that has numerous applications in mathematics. It is used to simplify expressions that involve the difference of squares, and it is a key concept in solving quadratic equations. In addition, the formula is used in various mathematical operations, such as factoring and simplifying expressions.

Real-World Applications of the Difference of Squares Formula

The difference of squares formula has numerous real-world applications in various fields, such as physics, engineering, and computer science. For example, in physics, the formula is used to calculate the energy of a system, while in engineering, it is used to design and optimize systems. In computer science, the formula is used in algorithms and data structures.

Conclusion

In conclusion, the expression $\sqrt{25^2 - 24^2}$ is a classic example of a difference of squares, which can be simplified using the formula $(a^2 - b^2) = (a + b)(a - b)$. The difference of squares formula is a fundamental concept in algebra that has numerous applications in mathematics and real-world applications in various fields. By understanding and applying the formula, we can simplify expressions and solve problems in a more efficient and effective manner.

Additional Examples

Here are some additional examples of the difference of squares formula:

• $\sqrt{36^2 - 35^2} = \sqrt{(36 + 35)(36 - 35)} = \sqrt{(71)(1)} = \sqrt{71}$
• $\sqrt{81^2 - 80^2} = \sqrt{(81 + 80)(81 - 80)} = \sqrt{(161)(1)} = \sqrt{161}$
• $\sqrt{49^2 - 48^2} = \sqrt{(49 + 48)(49 - 48)} = \sqrt{(97)(1)} = \sqrt{97}$

These examples demonstrate the power and versatility of the difference of squares formula in simplifying expressions and solving problems.

Final Thoughts

In conclusion, the expression $\sqrt{25^2 - 24^2}$ is a classic example of a difference of squares, which can be simplified using the formula $(a^2 - b^2) = (a + b)(a - b)$. The difference of squares formula is a fundamental concept in algebra that has numerous applications in mathematics and real-world applications in various fields. By understanding and applying the formula, we can simplify expressions and solve problems in a more efficient and effective manner.

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Introduction

In the previous article, we evaluated the expression $\sqrt{25^2 - 24^2}$ using the difference of squares formula. In this article, we will answer some frequently asked questions (FAQs) about evaluating this expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that states:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be applied to any two numbers $a$ and $b$, and it is used to simplify expressions that involve the difference of squares.

Q: How do I apply the difference of squares formula to evaluate $\sqrt{25^2 - 24^2}$?

A: To apply the difference of squares formula, you need to identify the values of $a$ and $b$ in the expression. In this case, $a = 25$ and $b = 24$. Then, you can use the formula to simplify the expression as follows:

$$\sqrt{25^2 - 24^2} = \sqrt{(25 + 24)(25 - 24)}$$

Using the formula, you can simplify the expression further by evaluating the square root:

$$\sqrt{(25 + 24)(25 - 24)} = \sqrt{(49)(1)}$$

Now, you can simplify the expression further by evaluating the square root:

$$\sqrt{(49)(1)} = \sqrt{49} = 7$$

Q: What is the significance of the difference of squares formula in mathematics?

A: The difference of squares formula is a fundamental concept in algebra that has numerous applications in mathematics. It is used to simplify expressions that involve the difference of squares, and it is a key concept in solving quadratic equations. In addition, the formula is used in various mathematical operations, such as factoring and simplifying expressions.

Q: How do I use the difference of squares formula in real-world applications?

A: The difference of squares formula has numerous real-world applications in various fields, such as physics, engineering, and computer science. For example, in physics, the formula is used to calculate the energy of a system, while in engineering, it is used to design and optimize systems. In computer science, the formula is used in algorithms and data structures.

Q: Can I use the difference of squares formula to evaluate other expressions?

A: Yes, you can use the difference of squares formula to evaluate other expressions that involve the difference of squares. For example, you can use the formula to evaluate the expression $\sqrt{36^2 - 35^2}$ as follows:

$$\sqrt{36^2 - 35^2} = \sqrt{(36 + 35)(36 - 35)} = \sqrt{(71)(1)} = \sqrt{71}$$

Q: What are some common mistakes to avoid when using the difference of squares formula?

A: Some common mistakes to avoid when using the difference of squares formula include:

• Not identifying the values of $a$ and $b$ in the expression
• Not applying the formula correctly
• Not simplifying the expression further after applying the formula
• Not checking the result for errors

Q: How do I check my result for errors?

A: To check your result for errors, you can use the following steps:

• Simplify the expression using the difference of squares formula
• Evaluate the square root of the simplified expression
• Check the result for errors by plugging it back into the original expression

Conclusion

In conclusion, the difference of squares formula is a fundamental concept in algebra that has numerous applications in mathematics and real-world applications in various fields. By understanding and applying the formula, you can simplify expressions and solve problems in a more efficient and effective manner.