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

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 $8 \sqrt{5} + 2 \sqrt{45}$.

Breaking Down the Expression

To simplify the expression $8 \sqrt{5} + 2 \sqrt{45}$, we need to break it down into its individual components. The first step is to simplify the radical expression $\sqrt{45}$. We can do this by factoring the number 45 into its prime factors.

Factoring the Number 45

The prime factorization of 45 is $3^2 \cdot 5$. Therefore, we can rewrite the radical expression $\sqrt{45}$ as $\sqrt{3^2 \cdot 5}$.

Simplifying the Radical Expression

Using the property of radicals that $\sqrt{a^2} = a$, we can simplify the radical expression $\sqrt{3^2 \cdot 5}$ as $3 \sqrt{5}$.

Substituting the Simplified Expression

Now that we have simplified the radical expression $\sqrt{45}$, we can substitute it back into the original expression $8 \sqrt{5} + 2 \sqrt{45}$. This gives us $8 \sqrt{5} + 2 (3 \sqrt{5})$.

Simplifying the Expression Further

Using the distributive property, we can simplify the expression $8 \sqrt{5} + 2 (3 \sqrt{5})$ as $8 \sqrt{5} + 6 \sqrt{5}$.

Combining Like Terms

The final step is to combine like terms. In this case, we have two terms with the same radical expression, $\sqrt{5}$. We can combine them by adding their coefficients, which gives us $(8 + 6) \sqrt{5}$.

Simplifying the Final Expression

The final expression is $(8 + 6) \sqrt{5}$, which simplifies to $14 \sqrt{5}$.

Conclusion

In this article, we simplified the radical expression $8 \sqrt{5} + 2 \sqrt{45}$ by breaking it down into its individual components, factoring the number 45, simplifying the radical expression, substituting the simplified expression back into the original expression, simplifying the expression further, combining like terms, and simplifying the final expression. The final answer is $14 \sqrt{5}$.

Answer

The correct answer is C. $14 \sqrt{5}$.

Additional Tips and Tricks

• When simplifying radical expressions, it's essential to factor the number inside the radical sign into its prime factors.
• Use the property of radicals that $\sqrt{a^2} = a$ to simplify radical expressions.
• When combining like terms, add the coefficients of the terms with the same radical expression.
• Simplify the final expression by combining like terms and simplifying the radical expression.

Common Mistakes to Avoid

• Failing to factor the number inside the radical sign into its prime factors.
• Not using the property of radicals that $\sqrt{a^2} = a$ to simplify radical expressions.
• Not combining like terms when simplifying the expression.
• Not simplifying the final expression by combining like terms and simplifying the radical expression.

Real-World Applications

Radical expressions are used in various real-world applications, such as:

• Physics: Radical expressions are used to represent the square root of a number, which is essential in calculating distances, velocities, and accelerations.
• Engineering: Radical expressions are used to represent the square root of a number, which is essential in calculating stresses, strains, and loads.
• Computer Science: Radical expressions are used to represent the square root of a number, which is essential in calculating distances, velocities, and accelerations in computer graphics and game development.

Conclusion

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 follow these steps:

1. Factor the number inside the radical sign into its prime factors.
2. Use the property of radicals that $\sqrt{a^2} = a$ to simplify the radical expression.
3. Combine like terms by adding the coefficients of the terms with the same radical expression.
4. Simplify the final expression by combining like terms and simplifying the radical expression.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign, while an exponential expression is a mathematical expression that contains a power or an exponent. For example, $\sqrt{5}$ is a radical expression, while $5^2$ is an exponential expression.

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:

1. Factor the number inside the radical sign into its prime factors.
2. Use the property of radicals that $\sqrt{a^2} = a$ to simplify the radical expression.
3. Combine like terms by adding the coefficients of the terms with the same radical expression.
4. Simplify the final expression by combining like terms and simplifying the radical expression.

Q: What is the difference between a rational expression and a radical expression?

A: A rational expression is a mathematical expression that contains a fraction, while a radical expression is a mathematical expression that contains a root or a radical sign. For example, $\frac{1}{2}$ is a rational expression, while $\sqrt{5}$ is a radical expression.

Q: How do I simplify a radical expression with a rational coefficient?

A: To simplify a radical expression with a rational coefficient, you need to follow these steps:

1. Factor the number inside the radical sign into its prime factors.
2. Use the property of radicals that $\sqrt{a^2} = a$ to simplify the radical expression.
3. Simplify the rational coefficient by dividing it by the greatest common divisor of the numerator and denominator.
4. Combine like terms by adding the coefficients of the terms with the same radical expression.
5. Simplify the final expression by combining like terms and simplifying the radical expression.

Q: What is the difference between a radical expression and an absolute value expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign, while an absolute value expression is a mathematical expression that contains the absolute value of a number. For example, $\sqrt{5}$ is a radical expression, while $|5|$ is an absolute value expression.

Q: How do I simplify a radical expression with an absolute value coefficient?

A: To simplify a radical expression with an absolute value coefficient, you need to follow these steps:

1. Simplify the absolute value coefficient by evaluating the absolute value of the number.
2. Factor the number inside the radical sign into its prime factors.
3. Use the property of radicals that $\sqrt{a^2} = a$ to simplify the radical expression.
4. Combine like terms by adding the coefficients of the terms with the same radical expression.
5. Simplify the final expression by combining like terms and simplifying the radical expression.

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

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

• Failing to factor the number inside the radical sign into its prime factors.
• Not using the property of radicals that $\sqrt{a^2} = a$ to simplify the radical expression.
• Not combining like terms when simplifying the expression.
• Not simplifying the final expression by combining like terms and simplifying the radical expression.

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 radical expression using the steps outlined above.
2. Evaluate the simplified expression to ensure that it is correct.
3. Check that the simplified expression is in its simplest form by ensuring that there are no like terms that can be combined.
4. Check that the simplified expression is consistent with the original expression.

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

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

• Physics: Radical expressions are used to represent the square root of a number, which is essential in calculating distances, velocities, and accelerations.
• Engineering: Radical expressions are used to represent the square root of a number, which is essential in calculating stresses, strains, and loads.
• Computer Science: Radical expressions are used to represent the square root of a number, which is essential in calculating distances, velocities, and accelerations in computer graphics and game development.

Conclusion

In conclusion, simplifying radical expressions is an essential skill in mathematics, and it has various real-world applications. By following the steps outlined in this article, you can simplify radical expressions and solve problems involving radical expressions.