Which Choice Is Equivalent To The Expression Below?${ 5 \sqrt{10} + \sqrt{40} + \sqrt{90} }$A. ${ 13 \sqrt{10} }$B. ${ 10 \sqrt{10} }$C. ${ 18 \sqrt{10} }$D. ${ 7 \sqrt{10} }$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression: $5 \sqrt{10} + \sqrt{40} + \sqrt{90}$. We will examine each option 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 other root. 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 the Given Expression

To simplify the given expression, we need to first simplify each radical term. We can do this by factoring out perfect squares from the radicand (the number inside the square root).

Simplifying $\sqrt{40}$

We can simplify $\sqrt{40}$ by factoring out a perfect square:

$$\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$$

Simplifying $\sqrt{90}$

We can simplify $\sqrt{90}$ by factoring out a perfect square:

$$\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}$$

Simplifying the Entire Expression

Now that we have simplified each radical term, we can combine them to get the simplified expression:

$$5\sqrt{10} + 2\sqrt{10} + 3\sqrt{10} = (5 + 2 + 3)\sqrt{10} = 10\sqrt{10}$$

Evaluating the Options

Now that we have simplified the given expression, we can evaluate each option to determine which one is equivalent.

Option A: $13\sqrt{10}$

This option is not equivalent to the simplified expression, as the coefficient is different.

Option B: $10\sqrt{10}$

This option is equivalent to the simplified expression, as the coefficient and the radicand are the same.

Option C: $18\sqrt{10}$

This option is not equivalent to the simplified expression, as the coefficient is different.

Option D: $7\sqrt{10}$

This option is not equivalent to the simplified expression, as the coefficient is different.

Conclusion

In conclusion, the correct answer is Option B: $10\sqrt{10}$, as it is equivalent to the simplified expression. Simplifying radical expressions is a crucial skill in mathematics, and this article has provided a step-by-step guide on how to simplify the given expression. By following these steps, you can simplify any radical expression and determine which option is equivalent.

Additional Tips and Resources

• To simplify a radical expression, first simplify each radical term by factoring out perfect squares.
• Use the distributive property to combine like terms.
• Check your work by plugging in values or using a calculator.
• Practice simplifying radical expressions to become more comfortable with the process.

Common Mistakes to Avoid

• Failing to simplify each radical term before combining them.
• Not using the distributive property to combine like terms.
• Not checking your work by plugging in values or using a calculator.
• Not practicing simplifying radical expressions to become more comfortable with the process.

Real-World Applications

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

• Calculating distances and heights in geometry and trigonometry.
• Solving equations and inequalities in algebra and calculus.
• Working with electrical circuits and electronics.
• Designing and building structures, such as bridges and buildings.

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression: $5 \sqrt{10} + \sqrt{40} + \sqrt{90}$. We determined that the correct answer is Option B: $10\sqrt{10}$, as it is equivalent to the simplified expression. In this article, we will provide a Q&A guide to help you better understand and apply 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 other root. 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, first simplify each radical term by factoring out perfect squares. Use the distributive property to combine like terms. Check your work by plugging in values or using a calculator.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer and itself. For example, 4 is a perfect square because it can be expressed as 2 x 2.

Q: How do I factor out perfect squares?

A: To factor out perfect squares, look for numbers that can be expressed as the product of an integer and itself. For example, $\sqrt{40}$ can be factored out as $\sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to multiply a single value by multiple values. For example, $a(b + c) = ab + ac$.

Q: How do I use the distributive property to combine like terms?

A: To use the distributive property to combine like terms, multiply each value by the single value. For example, $(5 + 2 + 3)\sqrt{10} = 10\sqrt{10}$.

Q: Why is it important to check my work?

A: It is essential to check your work to ensure that you have simplified the radical expression correctly. You can do this by plugging in values or using a calculator.

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 simplify each radical term before combining them.
• Not using the distributive property to combine like terms.
• Not checking your work by plugging in values or using a calculator.
• Not practicing simplifying radical expressions to become more comfortable with the process.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises. You can also use online resources, such as calculators and worksheets, to help you practice.

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

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

• Calculating distances and heights in geometry and trigonometry.
• Solving equations and inequalities in algebra and calculus.
• Working with electrical circuits and electronics.
• Designing and building structures, such as bridges and buildings.

By mastering the skill of simplifying radical expressions, you can apply it to a wide range of real-world problems and become a more confident and competent mathematician.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can simplify any radical expression and determine which option is equivalent. Remember to practice simplifying radical expressions to become more comfortable with the process, and don't hesitate to ask for help if you need it.