Which Expression Is Equivalent To The Given Expression 45 \sqrt{45} 45 ​ ?A. 5 3 5 \sqrt{3} 5 3 ​ B. 5 9 5 \sqrt{9} 5 9 ​ C. 3 5 3 \sqrt{5} 3 5 ​ D. 9 5 9 \sqrt{5} 9 5 ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{45}$. We will examine the different options provided and determine which one is equivalent to the given expression.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. 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.

Breaking Down the Given Expression

The given expression is $\sqrt{45}$. To simplify this expression, we need to find the largest perfect square that divides 45. 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 squared.

Finding the Largest Perfect Square

To find the largest perfect square that divides 45, we can start by listing the factors of 45. The factors of 45 are 1, 3, 5, 9, 15, and 45. We can see that 9 is a perfect square because it can be expressed as 3 squared.

Simplifying the Expression

Now that we have found the largest perfect square that divides 45, we can simplify the expression. We can rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$. Since 9 is a perfect square, we can simplify the expression to $3\sqrt{5}$.

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options provided. The options are:

A. $5 \sqrt{3}$ B. $5 \sqrt{9}$ C. $3 \sqrt{5}$ D. $9 \sqrt{5}$

We can see that option C, $3 \sqrt{5}$, is equivalent to the simplified expression.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students to master. By breaking down the given expression and finding the largest perfect square that divides it, we can simplify the expression and determine which option is equivalent to it. In this case, the correct answer is option C, $3 \sqrt{5}$.

Tips and Tricks

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

• Look for perfect squares: A perfect square is a number that can be expressed as the square of an integer. Look for perfect squares that divide the number inside the square root.
• Break down the expression: Break down the expression into smaller parts and simplify each part separately.
• Use the properties of radicals: The properties of radicals state that the square root of a product is equal to the product of the square roots, and the square root of a quotient is equal to the quotient of the square roots.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying radical expressions:

• Not looking for perfect squares: Failing to look for perfect squares that divide the number inside the square root can lead to incorrect simplifications.
• Not breaking down the expression: Failing to break down the expression into smaller parts can lead to incorrect simplifications.
• Not using the properties of radicals: Failing to use the properties of radicals can lead to incorrect simplifications.

Real-World Applications

Simplifying radical expressions has many real-world applications. Here are a few examples:

• Engineering: Simplifying radical expressions is crucial in engineering, where it is used to calculate stresses and strains on structures.
• Physics: Simplifying radical expressions is crucial in physics, where it is used to calculate distances and velocities.
• Computer Science: Simplifying radical expressions is crucial in computer science, where it is used to calculate complex algorithms.

Conclusion

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Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\sqrt{45}$. We examined the different options provided and determined which one is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a 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 rewrite the expression as the product of the square root 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 square of an integer. For example, 16 is a perfect square because it can be expressed as 4 squared.

Q: How do I find the largest perfect square that divides a number?

A: To find the largest perfect square that divides a number, you can start by listing the factors of the number. You can then identify the perfect squares among the factors and choose the largest one.

Q: Can I simplify a radical expression with a negative number inside the square root?

A: Yes, you can simplify a radical expression with a negative number inside the square root. However, you need to remember that the square root of a negative number is an imaginary number.

Q: How do I simplify a radical expression with a variable inside the square root?

A: To simplify a radical expression with a variable inside the square root, you need to find the largest perfect square that divides the variable. You can then rewrite the expression as the product of the square root of the perfect square and the remaining variable.

Q: Can I simplify a radical expression with a fraction inside the square root?

A: Yes, you can simplify a radical expression with a fraction inside the square root. However, you need to remember that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

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

A: To simplify a radical expression with multiple square roots, you need to find the largest perfect square that divides each of the numbers inside the square roots. You can then rewrite the expression as the product of the square roots of the perfect squares and the remaining numbers.

Q: Can I simplify a radical expression with a radical inside another radical?

A: Yes, you can simplify a radical expression with a radical inside another radical. However, you need to remember that the square root of a radical is equal to the radical raised to the power of 1/2.

Q: How do I check if my simplified radical expression is correct?

A: To check if your simplified radical expression is correct, you can multiply the expression by itself to see if it equals the original expression.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students to master. By following the steps outlined in this Q&A guide, you can simplify radical expressions with ease and confidence. Remember to always look for perfect squares, break down the expression, and use the properties of radicals to simplify the expression.

Tips and Tricks

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

• Use a calculator: A calculator can help you find the square root of a number quickly and easily.
• Use a table of perfect squares: A table of perfect squares can help you quickly identify the perfect squares that divide a number.
• Use the properties of radicals: The properties of radicals state that the square root of a product is equal to the product of the square roots, and the square root of a quotient is equal to the quotient of the square roots.
• Check your work: Always check your work to make sure that your simplified radical expression is correct.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying radical expressions:

• Not looking for perfect squares: Failing to look for perfect squares that divide the number inside the square root can lead to incorrect simplifications.
• Not breaking down the expression: Failing to break down the expression into smaller parts can lead to incorrect simplifications.
• Not using the properties of radicals: Failing to use the properties of radicals can lead to incorrect simplifications.

Real-World Applications

Simplifying radical expressions has many real-world applications. Here are a few examples:

• Engineering: Simplifying radical expressions is crucial in engineering, where it is used to calculate stresses and strains on structures.
• Physics: Simplifying radical expressions is crucial in physics, where it is used to calculate distances and velocities.
• Computer Science: Simplifying radical expressions is crucial in computer science, where it is used to calculate complex algorithms.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students to master. By following the steps outlined in this Q&A guide, you can simplify radical expressions with ease and confidence. Remember to always look for perfect squares, break down the expression, and use the properties of radicals to simplify the expression.