To simplify the square root of 68, we need to express it in its simplest radical form by factoring out any perfect square factors.
Step‑by‑Step Simplification
- Identify the number – The radicand (the number inside the root) is 68.
- Factor the radicand – Write 68 as a product of two integers, one of which is a perfect square. The largest perfect square that divides 68 is 4, because 4 × 17 = 68.
- Apply the product property of square roots – Use the rule √(a × b) = √a × √b. This gives √68 = √(4 × 17) = √4 × √17.
- Simplify the perfect‑square root – Since √4 = 2, the expression becomes 2 × √17.
- Write the final simplified form – The square root of 68 simplifies to 2√17.
This sequence shows how the original radical is reduced to a product of an integer and a square‑root of a prime number, which cannot be simplified further.
Why Factoring Works
The ability to simplify √68 relies on the product property of radicals, which states that the square root of a product equals the product of the square roots, provided each factor is non‑negative. By splitting 68 into 4 (a perfect square) and 17 (a prime), we isolate the part that can be taken out of the radical. The remaining √17 stays inside because 17 has no square factors other than 1 Most people skip this — try not to. Nothing fancy..
Mathematically, if n = a × b where a is a perfect square, then √n = √a × √b = √a
