Simplify square root of 50
The Correct Answer and Explanation is :
To simplify the square root of 50, we start by breaking it down into its prime factors.
- Prime Factorization: The number 50 can be factored into prime numbers:
[
50 = 2 \times 5^2
]
Here, we see that 50 consists of the prime number 2 and the prime number 5 raised to the power of 2. - Using the Product Property of Square Roots: The square root of a product can be expressed as the product of the square roots:
[
\sqrt{50} = \sqrt{2 \times 5^2}
]
This allows us to separate the square root into two parts. - Simplifying Further: We can apply the square root to the factors:
[
\sqrt{50} = \sqrt{2} \times \sqrt{5^2}
]
Since the square root of (5^2) is simply 5, we can simplify further:
[
\sqrt{50} = \sqrt{2} \times 5
] - Final Simplification: Rearranging gives us:
[
\sqrt{50} = 5\sqrt{2}
]
Thus, the simplified form of (\sqrt{50}) is (5\sqrt{2}).
Explanation
The process of simplifying square roots involves recognizing perfect squares within the number under the square root. A perfect square is a number that can be expressed as the square of an integer. In this case, (5^2) (which is 25) is a perfect square that fits within 50. By separating the factors, we can simplify the square root into a more manageable expression.
This technique is particularly useful in mathematics, especially when dealing with equations or calculations that require simplified forms for ease of computation or when working with radical expressions in algebra.
Knowing how to simplify square roots can help in solving various problems, including those involving geometry, physics, and engineering, where exact values or further calculations are often required. The final answer, (5\sqrt{2}), indicates that while (\sqrt{50}) is not a whole number, it can be expressed in a simpler form that is easier to work with in mathematical contexts.