112 square root simplified
Understand the Problem
The question is asking how to simplify the square root of 112. This requires us to break down the number 112 into its prime factors and see if we can simplify the square root by expressing it in terms of integers.
Answer
The simplified form of $\sqrt{112}$ is $4\sqrt{7}$.
Answer for screen readers
The simplified form of the square root of 112 is $4\sqrt{7}$.
Steps to Solve
- Find the Prime Factors of 112
To simplify the square root, we first find the prime factors of 112. We can start dividing by the smallest prime number, which is 2.
$$ 112 \div 2 = 56 $$
Now, let's divide 56 by 2 again.
$$ 56 \div 2 = 28 $$
Continue dividing by 2.
$$ 28 \div 2 = 14 $$
And again.
$$ 14 \div 2 = 7 $$
Since 7 is a prime number, we stop here. The prime factors of 112 are: $$ 112 = 2^4 \times 7 $$
- Rewrite the Square Root Using Prime Factors
Now, we express the square root of 112 using its prime factors.
$$ \sqrt{112} = \sqrt{2^4 \times 7} $$
- Simplify the Square Root
Using the property of square roots, we can separate the square root of the product.
$$ \sqrt{112} = \sqrt{2^4} \times \sqrt{7} $$
Calculate $\sqrt{2^4}$, which is $2^2 = 4$. Therefore, we can simplify it to:
$$ \sqrt{112} = 4 \times \sqrt{7} $$
- Final Answer
The simplified form of the square root is:
$$ \sqrt{112} = 4\sqrt{7} $$
The simplified form of the square root of 112 is $4\sqrt{7}$.
More Information
The result $4\sqrt{7}$ shows how we can express square roots in a simpler form by using prime factorization. This technique is useful in various mathematical fields, including algebra and geometry.
Tips
- Forgetting to factor completely: Ensure all prime factors are found before simplifying.
- Misapplying square root properties: Remember that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ is key in simplifying products.
