Simplify the square root of 72.

Step-by-step solution

1. 1

   Look for the largest perfect-square factor. The perfect squares up to $72$ are $4,9,16,25,36,49,64$. Checking downward, $36$ divides $72$:

   $72=36\times 2$

   Starting from the largest square saves work; using $4$ first would give $2\sqrt{18}$ and require a second round.

2. 2

   Split the radical over the product. For non-negative $a,b$, $\sqrt{ab}=\sqrt a\sqrt b$:

   $\sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\cdot\sqrt2$

3. 3

   Evaluate the perfect-square part.

   $\sqrt{36}=6\quad\Longrightarrow\quad\sqrt{72}=6\sqrt2$

4. 4

   Confirm the radicand is now square-free. $2$ is prime, so no square factor remains and the expression is fully simplified. Equivalently via prime factors: $72=2^3\cdot 3^2$, and each pair of equal primes contributes one factor outside — a $3$ from $3^2$ and a $2$ from one pair of $2$s gives $6$, leaving a single $2$ inside.

5. 5

   Check numerically. $6\sqrt2\approx 6\times 1.4142136=8.4852814$, and squaring gives $72.000$ $\checkmark$.