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The square root property is a powerful tool in algebra, especially for solving quadratic equations. When you have an equation of the form \[ (x - a)^2 = b \], you can apply the square root property to both sides. This means taking the square root of both sides, which gives you: \[ (x - a) = \pm \sqrt{b} \]. This property comes in handy because it helps to simplify the equation and make it easier to solve. However, remember that when you take the square root of a number, you must consider both the positive and negative roots.

Simplifying radicals is crucial when solving equations that involve square roots. After you take the square root of both sides of the equation, you may need to simplify the radical expression. For instance, in the equation \[ \sqrt{27} \], you can simplify it because 27 can be factored into 9 and 3. Since 9 is a perfect square, it can be simplified as follows: \[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3} \]. Breaking the numbers down into their prime factors or recognizing perfect squares will help you simplify.

Solving quadratic equations step-by-step involves breaking down the problem into more manageable stages. Let’s illustrate with the provided exercise: \((x-8)^{2}=27\). \The first step is to isolate the squared term, which in this case is already done because \( (x-8)^2 = 27 \).Apply the square root property to both sides: \( \sqrt{(x-8)^2} = \sqrt{27} \). Now, the equation becomes \( x - 8 = \pm \sqrt{27} \).Next, simplify the radical \(\sqrt{27} = 3 \sqrt{3} \).Hence, the equation simplifies to: \( x - 8 = \pm 3 \sqrt{3} \). The final step is to isolate \x\ by adding 8 to both sides: \( x = 8 \pm 3 \sqrt{3} \). Consequently, the solutions are: \( x = 8 + 3 \sqrt{3} \) and \( x = 8 - 3 \sqrt{3} \).