Simplify The Two Terms Into One Radical Expression: $4 \sqrt{250} - 10 \sqrt{10}$Show Your Work Here:Hint: To Add The Square Root Symbol, Type root

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will simplify the given expression $4 \sqrt{250} - 10 \sqrt{10}$ into one radical expression. We will break down the steps involved in simplifying radical expressions and provide a clear, step-by-step guide.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root 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.

Step 1: Simplify the First Term

The first term in the given expression is $4 \sqrt{250}$. To simplify this term, we need to find the prime factorization of 250.

Prime Factorization of 250

The prime factorization of 250 is:

250 = 2 × 125

Since 125 is a perfect square (11 × 11), we can rewrite 250 as:

250 = 2 × 11 × 11

Now, we can rewrite the first term as:

$4 \sqrt{250} = 4 \sqrt{2 × 11 × 11}$

Simplifying the First Term

We can simplify the first term by taking the square root of the perfect square (11 × 11):

$4 \sqrt{2 × 11 × 11} = 4 \sqrt{2} × \sqrt{11 × 11}$

$4 \sqrt{2} × \sqrt{11 × 11} = 4 \sqrt{2} × 11$

$4 \sqrt{2} × 11 = 44 \sqrt{2}$

So, the simplified first term is $44 \sqrt{2}$.

Step 2: Simplify the Second Term

The second term in the given expression is $-10 \sqrt{10}$. To simplify this term, we need to find the prime factorization of 10.

Prime Factorization of 10

The prime factorization of 10 is:

10 = 2 × 5

Now, we can rewrite the second term as:

$-10 \sqrt{10} = -10 \sqrt{2 × 5}$

Simplifying the Second Term

We can simplify the second term by taking the square root of the perfect square (2 × 5):

$-10 \sqrt{2 × 5} = -10 \sqrt{2} × \sqrt{5}$

$-10 \sqrt{2} × \sqrt{5} = -10 \sqrt{10}$

So, the simplified second term is $-10 \sqrt{10}$.

Step 3: Combine the Simplified Terms

Now that we have simplified both terms, we can combine them to get the final simplified expression.

Combining the Simplified Terms

$44 \sqrt{2} - 10 \sqrt{10}$

We can rewrite the expression as:

$44 \sqrt{2} - 10 \sqrt{10} = 44 \sqrt{2} - 10 \sqrt{2 × 5}$

$44 \sqrt{2} - 10 \sqrt{2 × 5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

$44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5} = 44 \sqrt{2} - 10 \sqrt{2} × \sqrt{5}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will provide a Q&A guide to help you understand and simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the prime factorization of the number inside the square root. Then, you can rewrite the expression using the prime factorization and simplify it further.

Q: What is the prime factorization of a number?

A: The prime factorization of a number is the expression of the number as a product of prime numbers. For example, the prime factorization of 250 is 2 × 125, and the prime factorization of 125 is 11 × 11.

Q: How do I simplify a radical expression with a perfect square?

A: If the number inside the square root is a perfect square, you can rewrite the expression using the prime factorization and simplify it further. For example, the expression $4 \sqrt{250}$ can be simplified as $4 \sqrt{2 × 11 × 11}$, which is equal to $4 \sqrt{2} × \sqrt{11 × 11}$.

Q: How do I simplify a radical expression with a non-perfect square?

A: If the number inside the square root is not a perfect square, you cannot simplify the expression further. For example, the expression $4 \sqrt{250}$ cannot be simplified further because 250 is not a perfect square.

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

A: Yes, you can simplify a radical expression with multiple terms by simplifying each term separately and then combining them. For example, the expression $4 \sqrt{250} - 10 \sqrt{10}$ can be simplified as $44 \sqrt{2} - 10 \sqrt{10}$.

Q: How do I know if a radical expression is simplified?

A: A radical expression is simplified if it cannot be simplified further. In other words, if the number inside the square root is a perfect square, you can simplify the expression further. If the number inside the square root is not a perfect square, you cannot simplify the expression further.

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 square root
• Not rewriting the expression using the prime factorization
• Not simplifying the expression further when possible
• Not combining multiple terms correctly

Conclusion

Simplifying radical expressions is a crucial skill to master in mathematics. By understanding the prime factorization of numbers and simplifying radical expressions, you can solve complex mathematical problems with ease. Remember to always find the prime factorization of the number inside the square root, rewrite the expression using the prime factorization, and simplify it further when possible.