To simplify \(\sqrt{450}\), we start by factoring \(450\) into its prime factors.
First, we can divide \(450\) by \(2\): \[ 450 \div 2 = 225 \] So, we have: \[ 450 = 2 \times 225 \]
Next, we can factor \(225\): \[ 225 \div 15 = 15 \] or \[ 225 = 15 \times 15 = 15^2 \] We can further factor \(15\): \[ 15 = 3 \times 5 \] Thus, we have: \[ 225 = 3^2 \times 5^2 \] Combining all these factors, we have: \[ 450 = 2 \times 3^2 \times 5^2 \]
Now we can rewrite the square root: \[ \sqrt{450} = \sqrt{2 \times 3^2 \times 5^2} \]
We can apply the property of square roots that allows us to separate the perfect squares: \[ \sqrt{450} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5^2} \] Since \(\sqrt{3^2} = 3\) and \(\sqrt{5^2} = 5\), this simplifies to: \[ \sqrt{450} = \sqrt{2} \times 3 \times 5 \]
Now multiplying \(3\) and \(5\): \[ 3 \times 5 = 15 \] Thus, we have: \[ \sqrt{450} = 15\sqrt{2} \]
Therefore, the simplified form is: \[ \boxed{15\sqrt{2}} \]