Final answer:
To simplify the expression √(27a) assuming a typo in the original question, factor out the perfect square from 27, which is 9, resulting in the simplified form 3√(3a). The same approach applies to any perfect square factors in 'a'. Always eliminate terms where possible and check the final answer.
Step-by-step explanation:
To simplify the expression 27a by removing all perfect squares from inside the square root, we should first identify any perfect square factors of 27a. Since 27 is 3³, we can't directly take a square root without an additional step. However, if the expression was written incorrectly and we presume it to be √(27a), we would proceed to factor 27 as 9 × 3, where 9 is a perfect square. Assuming 'a' is also a variable that may contain a perfect square, we would factor that out as well.
If 'a' does not contain any perfect square factors, then the expression √(27a) simplifies to 3√(3a), as 9 is the perfect square of 3 and can be taken out of the square root. If 'a' did contain perfect square factors, those would be similarly factored out. To simplify a square root expression, one needs to eliminate terms wherever possible. This process is similar to working with integer powers and fractional powers.
For example, x² can be expressed as √x. Therefore, the simplification process involves identifying perfect square factors of the radicand (the number under the square root), taking those factors out of the square root as integers, and then multiplying them by the square roots that remain. As a final step, always check the answer to ensure it is reasonable.