Simplifying Square Roots
A square root is simplified when no perfect-square factor remains inside the radical. The goal is to rewrite the root using the largest possible perfect square.
Method
- Factor the number inside the square root into a product that includes a perfect square.
- Split the radical: (\sqrt{ab} = \sqrt a\sqrt b), when both parts are nonnegative.
- Take the square root of the perfect square and move it outside the radical.
- Leave any remaining factor inside the square root.
Example pattern
If you have (\sqrt{48}), look for a perfect-square factor:
(48 = 16 \cdot 3), so
(\sqrt{48} = \sqrt{16\cdot 3} = 4\sqrt{3}).
Good checks
- The number inside the radical should have no square factors left.
- If possible, mentally verify by squaring the simplified form. For example, ((4\sqrt3)^2 = 16\cdot 3 = 48).
Tips
- Common perfect squares are (4, 9, 16, 25, 36, 49, 64, 81, 100).
- If the radicand is already a perfect square, the answer is a whole number.
- Keep coefficients outside the radical in simplest form.