What is the simplest form of the square root of 50?
Understand the Problem
The question is asking for the simplification of the square root of 50. This involves finding the prime factors of 50 and simplifying the radical expression accordingly.
Answer
$5\sqrt{2}$
Answer for screen readers
The simplified form of $\sqrt{50}$ is $5\sqrt{2}$.
Steps to Solve
- Find the prime factors of 50
First, we need to express 50 as a product of its prime factors.
50 can be factored as:
$$ 50 = 2 \times 25 $$
25 can be further factored:
$$ 25 = 5 \times 5 $$
So, the prime factorization of 50 is:
$$ 50 = 2 \times 5^2 $$
- Apply the property of square roots
Next, we can use the property of square roots that states:
$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$
We can apply this to the prime factors of 50:
$$ \sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2} $$
- Simplify the expression
Since the square root of a square returns the base:
$$ \sqrt{5^2} = 5 $$
This gives us:
$$ \sqrt{50} = 5 \times \sqrt{2} $$
So, the simplified form of $\sqrt{50}$ is:
$$ 5\sqrt{2} $$
The simplified form of $\sqrt{50}$ is $5\sqrt{2}$.
More Information
The simplification of radicals is a key skill in algebra. Recognizing and extracting square factors helps simplify expressions, making them easier to work with.
Tips
One common mistake is not recognizing that $5^2$ can be taken out of the square root, leading to the incorrect answer of just $\sqrt{50}$ instead of simplifying it to $5\sqrt{2}$. Another mistake could be failing to fully factor the number to its prime factors.
