Combine These Radicals: 8 5 + 2 45 8 \sqrt{5} + 2 \sqrt{45} 8 5 ​ + 2 45 ​ Options:A. 4 5 4 \sqrt{5} 4 5 ​ B. 10 5 10 \sqrt{5} 10 5 ​ C. 14 5 14 \sqrt{5} 14 5 ​ D. 74 45 74 \sqrt{45} 74 45 ​

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Introduction

Radicals, also known as roots, are an essential part of mathematics, particularly in algebra and geometry. They are used to represent the square root or other roots of a number. In this article, we will focus on simplifying radical expressions, which is a crucial skill in mathematics. We will use the given problem, 85+2458 \sqrt{5} + 2 \sqrt{45}, to demonstrate the steps involved in simplifying radical expressions.

Understanding Radicals

Before we dive into simplifying radical expressions, it's essential to understand what radicals are. A radical is a symbol used to represent the square root or other roots 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.

Simplifying Radical Expressions

Simplifying radical expressions involves combining like terms and using the properties of radicals to simplify the expression. Let's start by simplifying the given problem, 85+2458 \sqrt{5} + 2 \sqrt{45}.

Step 1: Simplify the Second Term

The second term, 2452 \sqrt{45}, can be simplified by finding the prime factorization of 45. The prime factorization of 45 is 3253^2 \cdot 5. We can rewrite the second term as 23252 \sqrt{3^2 \cdot 5}.

Step 2: Simplify the Second Term Further

Now that we have the prime factorization of 45, we can simplify the second term further. We can rewrite the second term as 2352 \cdot 3 \sqrt{5}, because the square root of 323^2 is 3.

Step 3: Combine Like Terms

Now that we have simplified the second term, we can combine like terms. The first term is 858 \sqrt{5}, and the simplified second term is 656 \sqrt{5}. We can combine these two terms by adding their coefficients.

Step 4: Simplify the Expression

Now that we have combined like terms, we can simplify the expression. The expression becomes (8+6)5(8 + 6) \sqrt{5}, which simplifies to 14514 \sqrt{5}.

Conclusion

In this article, we have demonstrated the steps involved in simplifying radical expressions. We used the given problem, 85+2458 \sqrt{5} + 2 \sqrt{45}, to show how to simplify radical expressions by combining like terms and using the properties of radicals. We simplified the second term by finding its prime factorization and then simplified it further by rewriting it in terms of the square root of 5. Finally, we combined like terms and simplified the expression to get the final answer, 14514 \sqrt{5}.

Answer

The final answer is C. 14514 \sqrt{5}.

Additional Tips and Tricks

  • When simplifying radical expressions, it's essential to find the prime factorization of the number inside the radical.
  • Use the properties of radicals, such as the product rule and the quotient rule, to simplify the expression.
  • Combine like terms by adding their coefficients.
  • Simplify the expression by rewriting it in terms of the square root of a perfect square.

Common Mistakes to Avoid

  • Not finding the prime factorization of the number inside the radical.
  • Not using the properties of radicals to simplify the expression.
  • Not combining like terms by adding their coefficients.
  • Not simplifying the expression by rewriting it in terms of the square root of a perfect square.

Real-World Applications

Simplifying radical expressions has many real-world applications, such as:

  • Calculating the area and perimeter of shapes with radical dimensions.
  • Finding the volume of shapes with radical dimensions.
  • Simplifying complex expressions in physics and engineering.
  • Solving problems in geometry and trigonometry.

Conclusion

Introduction

In our previous article, we demonstrated the steps involved in simplifying radical expressions. In this article, we will provide a Q&A guide to help you understand and apply the concepts of simplifying radical expressions.

Q: What is a radical?

A: A radical is a symbol used to represent the square root or other roots of a number. 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 follow these steps:

  1. Find the prime factorization of the number inside the radical.
  2. Use the properties of radicals, such as the product rule and the quotient rule, to simplify the expression.
  3. Combine like terms by adding their coefficients.
  4. Simplify the expression by rewriting it in terms of the square root of a perfect square.

Q: What is the product rule for radicals?

A: The product rule for radicals states that the product of two radicals is equal to the product of their coefficients multiplied by the radical of their product. In other words, ab=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Q: What is the quotient rule for radicals?

A: The quotient rule for radicals states that the quotient of two radicals is equal to the quotient of their coefficients divided by the radical of their quotient. In other words, ab=abbb=abb\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{ab}}{b}.

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 before. However, you may need to use the properties of radicals to simplify the expression.

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.

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

A: To simplify a radical expression with a fraction, you need to follow the same steps as before. However, you may need to use the properties of radicals to simplify the expression.

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

A: Yes, you can simplify a radical expression with a decimal. However, you need to remember that the decimal may not be a perfect square.

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

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

  • Not finding the prime factorization of the number inside the radical.
  • Not using the properties of radicals to simplify the expression.
  • Not combining like terms by adding their coefficients.
  • Not simplifying the expression by rewriting it in terms of the square root of a perfect square.

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

A: To check your work when simplifying radical expressions, you need to follow these steps:

  1. Simplify the expression using the steps outlined above.
  2. Check your work by plugging the simplified expression back into the original equation.
  3. Verify that the simplified expression is equal to the original expression.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify radical expressions and solve problems in mathematics. Remember to find the prime factorization of the number inside the radical, use the properties of radicals, combine like terms, and simplify the expression by rewriting it in terms of the square root of a perfect square.