Simplify The Expression: 112 \sqrt{112} 112 ​

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Introduction

Simplifying expressions involving square roots is an essential skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $\sqrt{112}$ using various techniques and strategies. We will explore the concept of prime factorization, identify perfect squares, and apply the properties of square roots to simplify the given expression.

Understanding Square Roots

Before we dive into simplifying the expression $\sqrt{112}$, let's briefly review the concept of square roots. A 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. We can represent square roots using the symbol $\sqrt{}$. For instance, $\sqrt{16}$ represents the square root of 16.

Prime Factorization

To simplify the expression $\sqrt{112}$, we need to find the prime factorization of 112. Prime factorization involves breaking down a number into its prime factors. A prime factor is a prime number that can be multiplied together to give the original number. For example, the prime factorization of 12 is $2^2 \times 3$, because 2 multiplied by 2 multiplied by 3 equals 12.

Finding the Prime Factorization of 112

To find the prime factorization of 112, we can start by dividing it by the smallest prime number, which is 2. We can write 112 as $2 \times 56$. Since 56 is also divisible by 2, we can write it as $2 \times 28$. We can continue this process until we reach a number that is not divisible by 2.

$112 = 2 \times 56$ $56 = 2 \times 28$ $28 = 2 \times 14$ $14 = 2 \times 7$

Therefore, the prime factorization of 112 is $2^4 \times 7$.

Identifying Perfect Squares

Now that we have the prime factorization of 112, we can identify the perfect squares. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. We can identify the perfect squares in the prime factorization of 112 by looking for pairs of the same prime factor.

Simplifying the Expression

Now that we have identified the perfect squares in the prime factorization of 112, we can simplify the expression $\sqrt{112}$. We can rewrite the expression as $\sqrt{2^4 \times 7}$. Since $2^4$ is a perfect square, we can simplify it as $2^2$. Therefore, the simplified expression is $\sqrt{2^2 \times 7}$.

Applying the Properties of Square Roots

We can further simplify the expression $\sqrt{2^2 \times 7}$ by applying the properties of square roots. One of the properties of square roots is that the square root of a product is equal to the product of the square roots. For example, $\sqrt{2 \times 3} = \sqrt{2} \times \sqrt{3}$. We can apply this property to simplify the expression $\sqrt{2^2 \times 7}$.

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

Therefore, the simplified expression is $2\sqrt{7}$.

Conclusion

In this article, we simplified the expression $\sqrt{112}$ using various techniques and strategies. We found the prime factorization of 112, identified the perfect squares, and applied the properties of square roots to simplify the expression. We learned that the simplified expression is $2\sqrt{7}$. This article demonstrates the importance of simplifying expressions involving square roots in mathematics, particularly in algebra and geometry.

Frequently Asked Questions

• What is the prime factorization of 112? The prime factorization of 112 is $2^4 \times 7$.
• How do you simplify the expression $\sqrt{112}$? To simplify the expression $\sqrt{112}$, we can find the prime factorization of 112, identify the perfect squares, and apply the properties of square roots.
• What is the simplified expression of $\sqrt{112}$? The simplified expression of $\sqrt{112}$ is $2\sqrt{7}$.

Final Thoughts

Simplifying expressions involving square roots is an essential skill in mathematics. By understanding the concept of prime factorization, identifying perfect squares, and applying the properties of square roots, we can simplify complex expressions and solve problems in algebra and geometry. In this article, we simplified the expression $\sqrt{112}$ using various techniques and strategies. We learned that the simplified expression is $2\sqrt{7}$. This article demonstrates the importance of simplifying expressions involving square roots in mathematics.

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Introduction

In our previous article, we simplified the expression $\sqrt{112}$ using various techniques and strategies. We found the prime factorization of 112, identified the perfect squares, and applied the properties of square roots to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying the expression $\sqrt{112}$.

Q&A

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 expression $\sqrt{112}$?

A: To simplify the expression $\sqrt{112}$, we can find the prime factorization of 112, identify the perfect squares, and apply the properties of square roots.

Q: What is the simplified expression of $\sqrt{112}$?

A: The simplified expression of $\sqrt{112}$ is $2\sqrt{7}$.

Q: Can you explain the concept of prime factorization?

A: Prime factorization is the process of breaking down a number into its prime factors. A prime factor is a prime number that can be multiplied together to give the original number. For example, the prime factorization of 12 is $2^2 \times 3$, because 2 multiplied by 2 multiplied by 3 equals 12.

Q: How do you identify perfect squares in the prime factorization of a number?

A: To identify perfect squares in the prime factorization of a number, we look for pairs of the same prime factor. For example, in the prime factorization of 112, we have $2^4$. Since 2 is a perfect square, we can simplify it as $2^2$.

Q: Can you explain the properties of square roots?

A: One of the properties of square roots is that the square root of a product is equal to the product of the square roots. For example, $\sqrt{2 \times 3} = \sqrt{2} \times \sqrt{3}$. We can apply this property to simplify the expression $\sqrt{2^2 \times 7}$.

Q: How do you apply the properties of square roots to simplify an expression?

A: To apply the properties of square roots to simplify an expression, we can use the property that the square root of a product is equal to the product of the square roots. For example, $\sqrt{2^2 \times 7} = \sqrt{2^2} \times \sqrt{7} = 2 \times \sqrt{7}$.

Q: Can you provide more examples of simplifying expressions involving square roots?

A: Yes, here are a few more examples:

• $\sqrt{16} = \sqrt{2^4} = 2^2 = 4$
• $\sqrt{25} = \sqrt{5^2} = 5$
• $\sqrt{36} = \sqrt{6^2} = 6$

Q: How do you simplify expressions involving square roots with negative numbers?

A: To simplify expressions involving square roots with negative numbers, we can use the property that the square root of a negative number is an imaginary number. For example, $\sqrt{-1} = i$, where $i$ is the imaginary unit.

Conclusion

In this article, we answered some frequently asked questions related to simplifying the expression $\sqrt{112}$. We explained the concept of prime factorization, identified perfect squares, and applied the properties of square roots to simplify the expression. We also provided more examples of simplifying expressions involving square roots and explained how to simplify expressions involving square roots with negative numbers.

Final Thoughts

Simplifying expressions involving square roots is an essential skill in mathematics. By understanding the concept of prime factorization, identifying perfect squares, and applying the properties of square roots, we can simplify complex expressions and solve problems in algebra and geometry. In this article, we answered some frequently asked questions related to simplifying the expression $\sqrt{112}$. We hope that this article has been helpful in understanding the concept of simplifying expressions involving square roots.