What is the square root of 224?
Understand the Problem
The question is asking for the square root of the number 224, which is a mathematical operation that requires finding a value that, when multiplied by itself, equals 224.
Answer
$4\sqrt{14}$
Answer for screen readers
The square root of 224 is $4\sqrt{14}$.
Steps to Solve
- Identify the number for which we need the square root
We are looking for the square root of 224, so we can express this as: $$ \sqrt{224} $$
- Simplify the square root if possible
We can factor 224 into its prime factors to simplify. First, we divide by 2: $$ 224 \div 2 = 112 $$
Continue factoring: $$ 112 \div 2 = 56 $$ $$ 56 \div 2 = 28 $$ $$ 28 \div 2 = 14 $$ $$ 14 \div 2 = 7 $$
So, the prime factorization of 224 is: $$ 224 = 2^5 \times 7 $$
- Use the property of square roots
Now we can simplify the square root using the property: $$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$
Using the factorization: $$ \sqrt{224} = \sqrt{2^5 \times 7} $$
We can extract squares from the square root: $$ \sqrt{224} = \sqrt{(2^4) \times (2^1) \times 7} = \sqrt{16} \times \sqrt{2 \times 7} = 4 \times \sqrt{14} $$
- Final result
Thus, our final answer for the square root of 224 is: $$ \sqrt{224} = 4\sqrt{14} $$
The square root of 224 is $4\sqrt{14}$.
More Information
The square root of 224 can be expressed in a simplified form. This shows how prime factorization can be used to simplify square roots. The number 14 itself is not a perfect square, which is why it remains under the square root.
Tips
- Miscalculating the prime factorization of 224 which can lead to an incorrect simplification.
- Forgetting to simplify further after finding the square root of the prime factors.
- Confusing the order of operations when extracting square roots.
