Simplify: $\sqrt{50} + \sqrt{18}$A. $34 \sqrt{2}$B. $5 \sqrt{2} + \sqrt{18}$C. $8 \sqrt{2}$D. $5 \sqrt{10} + 3$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given problem: $\sqrt{50} + \sqrt{18}$. We will break down the solution into manageable steps, using a combination of mathematical techniques and logical reasoning.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other root. 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. Radical expressions can be simplified by factoring out perfect squares, which are numbers that can be expressed as the product of a whole number and itself.

Simplifying $\sqrt{50}$

To simplify $\sqrt{50}$, we need to find the largest perfect square that divides 50. We can start by listing the factors of 50: 1, 2, 5, 10, 25, and 50. The largest perfect square that divides 50 is 25, which is equal to 5 multiplied by 5. Therefore, we can rewrite $\sqrt{50}$ as $\sqrt{25 \times 2}$.

Using the Property of Square Roots

The property of square roots states that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. We can apply this property to simplify $\sqrt{25 \times 2}$ as $\sqrt{25} \times \sqrt{2}$. Since $\sqrt{25}$ is equal to 5, we can further simplify the expression as $5 \times \sqrt{2}$.

Simplifying $\sqrt{18}$

To simplify $\sqrt{18}$, we need to find the largest perfect square that divides 18. We can start by listing the factors of 18: 1, 2, 3, 6, 9, and 18. The largest perfect square that divides 18 is 9, which is equal to 3 multiplied by 3. Therefore, we can rewrite $\sqrt{18}$ as $\sqrt{9 \times 2}$.

Using the Property of Square Roots Again

We can apply the property of square roots to simplify $\sqrt{9 \times 2}$ as $\sqrt{9} \times \sqrt{2}$. Since $\sqrt{9}$ is equal to 3, we can further simplify the expression as $3 \times \sqrt{2}$.

Combining the Simplified Expressions

Now that we have simplified both $\sqrt{50}$ and $\sqrt{18}$, we can combine the expressions to get the final result. We have $5 \times \sqrt{2} + 3 \times \sqrt{2}$.

Factoring Out the Common Term

We can factor out the common term $\sqrt{2}$ from both expressions. This gives us $(5 + 3) \times \sqrt{2}$.

Evaluating the Expression

Finally, we can evaluate the expression by multiplying the numbers inside the parentheses. This gives us $8 \times \sqrt{2}$.

Conclusion

In conclusion, the simplified expression for $\sqrt{50} + \sqrt{18}$ is $8 \sqrt{2}$. This result can be verified by plugging in the original values and simplifying the expression using the properties of square roots.

Final Answer

The final answer is $\boxed{8 \sqrt{2}}$.

Discussion

The given problem is a classic example of simplifying radical expressions. By breaking down the solution into manageable steps and using a combination of mathematical techniques and logical reasoning, we were able to simplify the expression and arrive at the final result. This problem requires a deep understanding of the properties of square roots and the ability to apply them in a practical context.

Related Problems

If you are looking for more practice problems on simplifying radical expressions, here are a few examples:

• $\sqrt{24} + \sqrt{32}$
• $\sqrt{27} + \sqrt{36}$
• $\sqrt{48} + \sqrt{64}$

These problems require the same level of mathematical sophistication and logical reasoning as the original problem. By practicing these problems, you can develop your skills and become more confident in your ability to simplify radical expressions.

Tips and Tricks

Here are a few tips and tricks to help you simplify radical expressions:

• Always look for perfect squares that divide the number inside the square root.
• Use the property of square roots to simplify the expression.
• Factor out common terms to simplify the expression.
• Evaluate the expression by multiplying the numbers inside the parentheses.

By following these tips and tricks, you can simplify radical expressions with ease and arrive at the final result with confidence.

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given problem: $\sqrt{50} + \sqrt{18}$. We will break down the solution into manageable steps, using a combination of mathematical techniques and logical reasoning.

Q&A: Simplifying Radical Expressions

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Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or other root. 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 find the largest perfect square that divides the number inside the square root. You can then use the property of square roots to simplify the expression.

Q: What is the property of square roots?

A: The property of square roots states that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. This means that you can simplify a radical expression by factoring out perfect squares.

Q: How do I factor out perfect squares?

A: To factor out perfect squares, you need to identify the largest perfect square that divides the number inside the square root. You can then rewrite the expression as the product 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 a whole number and itself. For example, 4 is a perfect square because it can be expressed as 2 multiplied by 2.

Q: How do I evaluate a simplified radical expression?

A: To evaluate a simplified radical expression, you need to multiply the numbers inside the parentheses. This will give you the final result.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not identifying the largest perfect square that divides the number inside the square root.
• Not using the property of square roots to simplify the expression.
• Not factoring out common terms.
• Not evaluating the expression correctly.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through example problems. You can also use online resources or math textbooks to find additional practice problems.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-world applications, including:

• Calculating distances and heights in geometry and trigonometry.
• Solving problems in physics and engineering.
• Working with financial data and statistics.
• Understanding and working with mathematical models in science and technology.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students and professionals alike. By understanding the properties of square roots and using mathematical techniques and logical reasoning, you can simplify radical expressions with ease. Remember to practice regularly and avoid common mistakes to become more confident in your ability to simplify radical expressions.

Final Tips

Here are a few final tips to help you simplify radical expressions:

• Always look for perfect squares that divide the number inside the square root.
• Use the property of square roots to simplify the expression.
• Factor out common terms to simplify the expression.
• Evaluate the expression by multiplying the numbers inside the parentheses.

By following these tips and practicing regularly, you can become a master of simplifying radical expressions and tackle even the most challenging problems with confidence.