Express in simplest radical form.
square root of, 10, end square root, square root of, 50, end square root
10
50
√10 * √50 = √(10 * 50) = √500 =
√(100 * 5) = √100 * √5 = 10√5
To express the given expression in simplest radical form, we need to simplify the square roots individually.
Let's start with the square root of 10:
1. Factorize the number inside the square root, which is 10. We can write 10 as the product of its prime factors: 10 = 2 × 5.
2. Simplify the square root by taking out any perfect square factors. In this case, there are no perfect square factors.
So, the square root of 10 remains the same: √10.
Now, let's simplify the square root of 50:
1. Factorize the number inside the square root, which is 50. We can write 50 as the product of its prime factors: 50 = 2 × 5 × 5.
2. Simplify the square root by taking out any perfect square factors. In this case, we have one perfect square factor, which is 5 × 5 = 25. So, we can rewrite 50 as 25 × 2.
3. Simplify further by taking the square root of the perfect square factor, which is √25 = 5.
Now, the square root of 50 becomes 5√2.
Putting it all together, the expression becomes:
√10 + √50 = √10 + 5√2.
To express the square roots of 10 and 50 in simplest radical form, we need to find the prime factors of the numbers inside the square roots.
Let's start with the square root of 10:
The prime factorization of 10 is 2 * 5.
Simplifying the square root of 10, we get √10 = √(2 * 5).
Now let's simplify the square root of 50:
The prime factorization of 50 is 2 * 5 * 5.
Simplifying the square root of 50, we get √50 = √(2 * 5 * 5).
Next, we can simplify the square roots by taking out pairs of the same prime factors:
For √10, we can take out one 5 from under the radical sign, so √10 = √(2 * 5) = √2√5.
For √50, we can take out two 5's from under the radical sign, so √50 = √(2 * 5 * 5) = √2√(5 * 5) = √2 * 5 = 5√2.
Therefore, the simplest radical form of √10 and √50 is: √10 = √2√5, and √50 = 5√2.