Adding Square Roots $$ \sqrt{50} + \sqrt{8} $$

Operations with Radicals: Adding Square Roots

In this lesson, we focus on operations with radicals, specifically adding square roots. Before radicals can be added, each square root must be simplified as much as possible. Only like radicals can be combined.

1. Theory of Adding Square Roots

Square roots can only be added together if they have the same radical part. This means the expression inside the square root must be identical. To achieve this, we often need to simplify each radical first.

2. Concept Explanation

Simplifying a square root involves factoring the number inside the radical into a perfect square multiplied by another number. The square root of the perfect square becomes a whole number, while the remaining factor stays inside the radical.

3. Formula or Steps

1. Simplify each square root separately.
2. Check that the radicals are the same.
3. Add the coefficients.
4. Write the final simplified result.

4. Example Problem

Problem:

$$ \sqrt{50} + \sqrt{8} $$

Solution:

Step 1: Simplify each square root.

$$ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} $$ $$ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} $$

Step 2: Add the like radical terms.

$$ 5\sqrt{2} + 2\sqrt{2} = (5 + 2)\sqrt{2} $$ $$ = 7\sqrt{2} $$

Final Answer:

$$ \sqrt{50} + \sqrt{8} = 7\sqrt{2} $$