Choosing the Equivalent Option

Oct 13, 2025 by miodeya.com 119 views

Understanding the Problem

The given expression is $\sqrt{40} + 2 \sqrt{10} + \sqrt{90}$ . We are asked to simplify this expression and choose the equivalent option from the given choices. To simplify the expression, we need to first break down each term into its prime factors and then combine like terms.

Breaking Down the Terms

Let's start by breaking down each term in the given expression:

$\sqrt{40}$ : We can break down 40 into its prime factors: $40 = 2^3 \cdot 5$ . Therefore, $\sqrt{40} = \sqrt{2^3 \cdot 5} = 2\sqrt{10}$ .
$\sqrt{90}$ : We can break down 90 into its prime factors: $90 = 2 \cdot 3^2 \cdot 5$ . Therefore, $\sqrt{90} = \sqrt{2 \cdot 3^2 \cdot 5} = 3\sqrt{10}$ .

Combining Like Terms

Now that we have broken down each term, we can combine like terms:

$\sqrt{40} + 2 \sqrt{10} + \sqrt{90} = 2\sqrt{10} + 2 \sqrt{10} + 3\sqrt{10} = 7\sqrt{10}$

Choosing the Equivalent Option

Based on our simplification, we can see that the given expression is equivalent to $7\sqrt{10}$ . Therefore, the correct answer is:

D. $7 \sqrt{10}$

Conclusion

In this article, we have simplified the given expression $\sqrt{40} + 2 \sqrt{10} + \sqrt{90}$ and chosen the equivalent option from the given choices. We have broken down each term into its prime factors and combined like terms to arrive at the simplified expression. The correct answer is $7\sqrt{10}$ .

Key Takeaways

To simplify a radical expression, we need to break down each term into its prime factors.
We can combine like terms by adding or subtracting the coefficients of the like terms.
The correct answer is $7\sqrt{10}$ .

Final Answer

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root of a number. It is denoted by the symbol $\sqrt{}$ .

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down each term into its prime factors and then combine like terms. You can use the following steps:

Break down each term into its prime factors.
Combine like terms by adding or subtracting the coefficients of the like terms.
Simplify the expression by canceling out any common factors.

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

A: A radical expression is an expression that contains a square root or other root of a number, while an exponential expression is an expression that contains a power of a number. For example, $\sqrt{16}$ is a radical expression, while $2^4$ is an exponential expression.

Q: Can I simplify a radical expression with a negative number inside the square root?

A: Yes, you can simplify a radical expression with a negative number inside the square root. To do this, you need to break down the negative number into its prime factors and then simplify the expression.

Q: How do I simplify a radical expression with a variable inside the square root?

A: To simplify a radical expression with a variable inside the square root, you need to break down the variable into its prime factors and then simplify the expression. You can use the following steps:

Break down the variable into its prime factors.
Simplify the expression by canceling out any common factors.
Write the simplified expression in the form $a\sqrt{b}$ , where $a$ and $b$ are integers.

Q: Can I simplify a radical expression with a fraction inside the square root?

A: Yes, you can simplify a radical expression with a fraction inside the square root. To do this, you need to break down the fraction into its prime factors and then simplify the expression.

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

A: To simplify a radical expression with multiple terms, you need to break down each term into its prime factors and then combine like terms. You can use the following steps:

Break down each term into its prime factors.
Combine like terms by adding or subtracting the coefficients of the like terms.
Simplify the expression by canceling out any common factors.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $\boxed{D. $7 \sqrt{10}$ }$.

Conclusion

In this article, we have answered some frequently asked questions on simplifying radical expressions. We have covered topics such as breaking down terms into prime factors, combining like terms, and simplifying expressions with variables and fractions inside the square root. We hope that this article has been helpful in understanding how to simplify radical expressions.