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

Understanding Radical Expressions

Radical expressions are mathematical expressions that involve the use of roots or radicals. In this article, we will focus on simplifying radical expressions, specifically the expression $\sqrt{45}$. We will explore the different options available and determine which one is equivalent to the given expression.

Breaking Down the Expression

To simplify the expression $\sqrt{45}$, we need to break it down into its prime factors. The prime factorization of 45 is $3^2 \cdot 5$. This means that we can rewrite the expression as $\sqrt{3^2 \cdot 5}$.

Simplifying the Expression

Now that we have broken down the expression into its prime factors, we can simplify it further. We can rewrite the expression as $\sqrt{3^2} \cdot \sqrt{5}$. Since the square root of $3^2$ is simply 3, we can simplify the expression to $3\sqrt{5}$.

Evaluating the Options

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

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

**Option A: $9 \sqrt{5}$

Option A is not equivalent to the simplified expression. The square root of 9 is 3, but the expression is multiplied by $\sqrt{5}$, which is not present in the simplified expression.

**Option B: $5 \sqrt{9}$

Option B is also not equivalent to the simplified expression. The square root of 9 is 3, but the expression is multiplied by $\sqrt{5}$, which is not present in the simplified expression.

**Option C: $3 \sqrt{5}$

Option C is equivalent to the simplified expression. The square root of 45 is $3\sqrt{5}$, which matches the simplified expression.

**Option D: $5 \sqrt{3}$

Option D is not equivalent to the simplified expression. The square root of 5 is not present in the simplified expression, and the expression is multiplied by $\sqrt{3}$, which is not present in the simplified expression.

Conclusion

In conclusion, the correct answer is option C: $3 \sqrt{5}$. This is equivalent to the simplified expression $\sqrt{45}$.

Tips and Tricks

When simplifying radical expressions, it is essential to break down the expression into its prime factors. This will help you identify the square root of each factor and simplify the expression further. Additionally, be sure to evaluate each option carefully to ensure that it matches the simplified expression.

Common Mistakes

When simplifying radical expressions, it is common to make mistakes such as:

• Not breaking down the expression into its prime factors
• Not identifying the square root of each factor
• Not evaluating each option carefully

To avoid these mistakes, be sure to follow the steps outlined in this article and take your time when simplifying radical expressions.

Real-World Applications

Simplifying radical expressions has numerous real-world applications, including:

• Calculating the area and perimeter of shapes with radical dimensions
• Solving equations involving radical expressions
• Working with mathematical models that involve radical expressions

By mastering the art of simplifying radical expressions, you will be better equipped to tackle complex mathematical problems and apply mathematical concepts to real-world situations.

Practice Problems

To practice simplifying radical expressions, try the following problems:

• Simplify the expression $\sqrt{24}$
• Simplify the expression $\sqrt{48}$
• Simplify the expression $\sqrt{72}$

By practicing these problems, you will become more comfortable with simplifying radical expressions and be better equipped to tackle complex mathematical problems.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that involves the use of roots or radicals. It is a way of expressing a number that is a product of a perfect square and a non-perfect square.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into its prime factors. This will help you identify the square root of each factor and simplify the expression further.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, such as 4, 9, 16, etc. A non-perfect square is a number that cannot be expressed as the square of an integer, such as 2, 3, 5, etc.

Q: How do I identify the square root of a factor?

A: To identify the square root of a factor, you need to look for the largest perfect square that divides the factor. For example, if you have the factor 24, you can break it down into its prime factors as 2^3 * 3. The largest perfect square that divides 24 is 4, which is the square root of 2^2.

Q: Can I simplify a radical expression with a negative number?

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

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to follow the same steps as you would with a numerical expression. However, you need to be careful when simplifying expressions with variables, as the variable may have different values in different contexts.

Q: Can I simplify a radical expression with a fraction?

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

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

A: To simplify a radical expression with multiple factors, you need to break down each factor into its prime factors and then simplify the expression further.

Q: Can I simplify a radical expression with a decimal number?

A: Yes, you can simplify a radical expression with a decimal number. However, you need to remember that the square root of a decimal number is an irrational number, which cannot be expressed as a finite decimal or fraction.

Q: How do I check my work when simplifying a radical expression?

A: To check your work when simplifying a radical expression, you need to multiply the simplified expression by itself to see if it equals the original expression.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill that requires practice and patience. By following the steps outlined in this article and practicing with real-world problems, you will become more confident in your ability to simplify radical expressions and tackle complex mathematical problems.

Practice Problems

To practice simplifying radical expressions, try the following problems:

• Simplify the expression $\sqrt{48}$
• Simplify the expression $\sqrt{72}$
• Simplify the expression $\sqrt{96}$

By practicing these problems, you will become more comfortable with simplifying radical expressions and be better equipped to tackle complex mathematical problems.

Common Mistakes

When simplifying radical expressions, it is common to make mistakes such as:

• Not breaking down the expression into its prime factors
• Not identifying the square root of each factor
• Not evaluating each option carefully

To avoid these mistakes, be sure to follow the steps outlined in this article and take your time when simplifying radical expressions.

Real-World Applications

Simplifying radical expressions has numerous real-world applications, including:

• Calculating the area and perimeter of shapes with radical dimensions
• Solving equations involving radical expressions
• Working with mathematical models that involve radical expressions

By mastering the art of simplifying radical expressions, you will be better equipped to tackle complex mathematical problems and apply mathematical concepts to real-world situations.