Perform The Indicated Operation And Simplify The Answer When Possible.$\[ 5 \sqrt{8} + 8 \sqrt{18} \\]

Understanding Radical Expressions

Radical expressions are mathematical expressions that contain a root or a radical sign. They are used to represent the square root or other roots of a number. In this article, we will focus on simplifying radical expressions, specifically the expression ${5 \sqrt{8} + 8 \sqrt{18}}$.

Breaking Down the Expression

To simplify the given expression, we need to break it down into its individual components. The expression consists of two terms: ${5 \sqrt{8}}$ and ${8 \sqrt{18}}$. We can simplify each term separately before combining them.

Simplifying the First Term

The first term is ${5 \sqrt{8}}$. To simplify this term, we need to find the prime factorization of 8. The prime factorization of 8 is ${2^3}$. We can rewrite the term as ${5 \sqrt{2^3}}$.

Using the Property of Radicals

We can use the property of radicals that states ${\sqrt{a^2} = a}$. In this case, we can rewrite ${\sqrt{2^3}}$ as ${\sqrt{2^2 \cdot 2}}$. This simplifies to ${2\sqrt{2}}$.

Simplifying the First Term (Continued)

Now that we have simplified the radical part of the first term, we can rewrite it as ${5 \cdot 2\sqrt{2}}$. This simplifies to ${10\sqrt{2}}$.

Simplifying the Second Term

The second term is ${8 \sqrt{18}}$. To simplify this term, we need to find the prime factorization of 18. The prime factorization of 18 is ${2 \cdot 3^2}$. We can rewrite the term as ${8 \sqrt{2 \cdot 3^2}}$.

Using the Property of Radicals (Again)

We can use the property of radicals that states ${\sqrt{a^2} = a}$. In this case, we can rewrite ${\sqrt{2 \cdot 3^2}}$ as ${\sqrt{2 \cdot 3 \cdot 3}}$. This simplifies to ${3\sqrt{2 \cdot 3}}$.

Simplifying the Second Term (Continued)

Now that we have simplified the radical part of the second term, we can rewrite it as ${8 \cdot 3\sqrt{2 \cdot 3}}$. This simplifies to ${24\sqrt{6}}$.

Combining the Terms

Now that we have simplified both terms, we can combine them to get the final expression. The expression becomes ${10\sqrt{2} + 24\sqrt{6}}$.

Final Answer

The final answer is ${10\sqrt{2} + 24\sqrt{6}}$.

Conclusion

Simplifying radical expressions can be a challenging task, but with the right techniques and strategies, it can be done. In this article, we used the property of radicals to simplify the given expression. We broke down the expression into its individual components, simplified each term separately, and then combined them to get the final expression. With practice and patience, you can become proficient in simplifying radical expressions and solving complex mathematical problems.

Tips and Tricks

• Always start by breaking down the expression into its individual components.
• Use the property of radicals to simplify the radical part of each term.
• Combine the simplified terms to get the final expression.
• Practice, practice, practice! The more you practice, the more comfortable you will become with simplifying radical expressions.

Common Mistakes to Avoid

• Not breaking down the expression into its individual components.
• Not using the property of radicals to simplify the radical part of each term.
• Not combining the simplified terms to get the final expression.
• Not practicing regularly to improve your skills.

Real-World Applications

Simplifying radical expressions has many real-world applications. For example, in engineering, architects use radical expressions to calculate the dimensions of buildings and bridges. In physics, scientists use radical expressions to calculate the energy of particles. In finance, investors use radical expressions to calculate the value of investments.

Final Thoughts

Simplifying radical expressions is an essential skill for anyone who wants to succeed in mathematics and science. With practice and patience, you can become proficient in simplifying radical expressions and solving complex mathematical problems. Remember to always break down the expression into its individual components, use the property of radicals to simplify the radical part of each term, and combine the simplified terms to get the final expression. With these tips and tricks, you will be well on your way to becoming a master of simplifying radical expressions.

Frequently Asked Questions

Radical expressions can be a challenging topic for many students. In this article, we will answer 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 contains a root or a radical sign. It is used to represent the square root or other roots of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into its individual components, simplify each term separately, and then combine them to get the final expression.

Q: What is the property of radicals?

A: The property of radicals states that ${\sqrt{a^2} = a}$. This means that if you have a radical expression in the form of ${\sqrt{a^2}}$, you can simplify it to just ${a}$.

Q: How do I simplify a term with a radical?

A: To simplify a term with a radical, you need to find the prime factorization of the number inside the radical sign. Then, you can use the property of radicals to simplify the radical part of the term.

Q: Can I simplify a radical expression with multiple terms?

A: Yes, you can simplify a radical expression with multiple terms. Just break down each term separately, simplify each term, and then combine them to get the final expression.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as a fraction, such as 3/4 or 22/7. An irrational number is a number that cannot be expressed as a fraction, such as the square root of 2 or the square root of 3.

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

A: Yes, you can simplify a radical expression with a rational number. Just multiply the rational number by the radical expression to get the final result.

Q: What is the final answer to the expression ${5 \sqrt{8} + 8 \sqrt{18}}$?

A: The final answer to the expression ${5 \sqrt{8} + 8 \sqrt{18}}$ is ${10\sqrt{2} + 24\sqrt{6}}$.

Q: Can I use a calculator to simplify a radical expression?

A: Yes, you can use a calculator to simplify a radical expression. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

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

A: Some common mistakes to avoid when simplifying radical expressions include not breaking down the expression into its individual components, not using the property of radicals to simplify the radical part of each term, and not combining the simplified terms to get the final expression.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying radical expressions on your own to see if you can get the correct answer.

Conclusion

Simplifying radical expressions can be a challenging task, but with practice and patience, you can become proficient in simplifying radical expressions and solving complex mathematical problems. Remember to always break down the expression into its individual components, use the property of radicals to simplify the radical part of each term, and combine the simplified terms to get the final expression. With these tips and tricks, you will be well on your way to becoming a master of simplifying radical expressions.

Final Thoughts

Simplifying radical expressions is an essential skill for anyone who wants to succeed in mathematics and science. With practice and patience, you can become proficient in simplifying radical expressions and solving complex mathematical problems. Remember to always check your work by hand to make sure you get the correct answer, and don't be afraid to ask for help if you need it.