Prime Factorization of 112

Oct 13, 2025 by miodeya.com 34 views

Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. It requires a deep understanding of the properties of square roots and the ability to manipulate expressions to their simplest form. In this article, we will simplify the given expression: $ -3 \sqrt{63} + 2 \sqrt{24} + 4 \sqrt{112} - \sqrt{3} $. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final simplified expression.

Breaking Down the Expression

The given expression consists of four terms, each involving a square root. To simplify the expression, we need to simplify each term separately.

Term 1: $ -3 \sqrt{63} $

The first term involves the square root of 63. To simplify this term, we need to find the prime factorization of 63.

Prime Factorization of 63

63 can be factored as: $ 63 = 3^2 \times 7 $

Now, we can rewrite the first term as:

$ -3 \sqrt{63} = -3 \sqrt{3^2 \times 7} $

Using the property of square roots, we can rewrite this as:

$ -3 \sqrt{63} = -3 \times 3 \sqrt{7} $

Simplifying further, we get:

$ -3 \sqrt{63} = -9 \sqrt{7} $

Term 2: $ 2 \sqrt{24} $

The second term involves the square root of 24. To simplify this term, we need to find the prime factorization of 24.

Prime Factorization of 24

24 can be factored as: $ 24 = 2^3 \times 3 $

Now, we can rewrite the second term as:

$ 2 \sqrt{24} = 2 \sqrt{2^3 \times 3} $

Using the property of square roots, we can rewrite this as:

$ 2 \sqrt{24} = 2 \times 2 \sqrt{3} $

Simplifying further, we get:

$ 2 \sqrt{24} = 4 \sqrt{3} $

Term 3: $ 4 \sqrt{112} $

The third term involves the square root of 112. To simplify this term, we need to find the prime factorization of 112.

Prime Factorization of 112

112 can be factored as: $ 112 = 2^4 \times 7 $

Now, we can rewrite the third term as:

$ 4 \sqrt{112} = 4 \sqrt{2^4 \times 7} $

Using the property of square roots, we can rewrite this as:

$ 4 \sqrt{112} = 4 \times 2 \sqrt{7} $

Simplifying further, we get:

$ 4 \sqrt{112} = 8 \sqrt{7} $

Term 4: $ - \sqrt{3} $

The fourth term involves the square root of 3. This term is already simplified.

Combining the Terms

Now that we have simplified each term, we can combine them to obtain the final simplified expression.

$ -3 \sqrt{63} + 2 \sqrt{24} + 4 \sqrt{112} - \sqrt{3} $

Substituting the simplified values of each term, we get:

$ -9 \sqrt{7} + 4 \sqrt{3} + 8 \sqrt{7} - \sqrt{3} $

Combining like terms, we get:

$ -9 \sqrt{7} + 8 \sqrt{7} + 4 \sqrt{3} - \sqrt{3} $

Simplifying further, we get:

$ - \sqrt{7} + 3 \sqrt{3} $

Therefore, the simplified expression is:

$ - \sqrt{7} + 3 \sqrt{3} $

Conclusion

Simplifying expressions involving square roots requires a deep understanding of the properties of square roots and the ability to manipulate expressions to their simplest form. In this article, we simplified the given expression: $ -3 \sqrt63} + 2 \sqrt{24} + 4 \sqrt{112} - \sqrt{3} $. We broke down the expression into smaller parts, simplified each part, and then combined them to obtain the final simplified expression. The simplified expression is $ - \sqrt{7 + 3 \sqrt{3} $.

Introduction

In our previous article, we simplified the given expression: $ -3 \sqrt{63} + 2 \sqrt{24} + 4 \sqrt{112} - \sqrt{3} $. We broke down the expression into smaller parts, simplified each part, and then combined them to obtain the final simplified expression. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the prime factorization of 63?

A: The prime factorization of 63 is $ 3^2 \times 7 $.

Q: How do you simplify the square root of 63?

A: To simplify the square root of 63, we can rewrite it as $ \sqrt{3^2 \times 7} $, which simplifies to $ 3 \sqrt{7} $.

Q: What is the prime factorization of 24?

A: The prime factorization of 24 is $ 2^3 \times 3 $.

Q: How do you simplify the square root of 24?

A: To simplify the square root of 24, we can rewrite it as $ \sqrt{2^3 \times 3} $, which simplifies to $ 2 \sqrt{3} $.

Q: What is the prime factorization of 112?

A: The prime factorization of 112 is $ 2^4 \times 7 $.

Q: How do you simplify the square root of 112?

A: To simplify the square root of 112, we can rewrite it as $ \sqrt{2^4 \times 7} $, which simplifies to $ 2^2 \sqrt{7} $ or $ 4 \sqrt{7} $.

Q: How do you combine like terms in the simplified expression?

A: To combine like terms in the simplified expression, we need to group the terms with the same variable. In this case, we have two terms with the variable $ \sqrt{7} $ and two terms with the variable $ \sqrt{3} $. We can combine the terms with the same variable by adding or subtracting their coefficients.

Q: What is the final simplified expression?

A: The final simplified expression is $ - \sqrt{7} + 3 \sqrt{3} $.

Tips and Tricks

Tip 1: Simplify the square root of each term separately

When simplifying an expression involving square roots, it's essential to simplify the square root of each term separately. This will make it easier to combine like terms later on.

Tip 2: Use the properties of square roots to simplify expressions

The properties of square roots can be used to simplify expressions. For example, we can rewrite $ \sqrt{a^2} $ as $ a $, and $ \sqrt{ab} $ as $ \sqrt{a} \times \sqrt{b} $.

Tip 3: Combine like terms carefully

When combining like terms, make sure to group the terms with the same variable. This will ensure that you get the correct simplified expression.

Conclusion

Simplifying expressions involving square roots requires a deep understanding of the properties of square roots and the ability to manipulate expressions to their simplest form. In this article, we answered some frequently asked questions related to the simplification of the expression: $ -3 \sqrt{63} + 2 \sqrt{24} + 4 \sqrt{112} - \sqrt{3} $. We provided tips and tricks to help you simplify expressions involving square roots.