Final answer:
The perfect squares are removed from the square root of √108a⁶ by factoring out the perfect square components, leaving the expression simplified as 6a³.
Step-by-step explanation:
The student's question asks to remove all perfect squares from inside the square root of the given expression √108a⁶, assuming a is positive. To do this, we should look for perfect square factors within 108 and a⁶ that can be taken outside the square root sign.
First, let's factor 108 into its prime factors: 108 = 2² × 3³. Notice that 2² (or 4) is a perfect square that can be removed from under the square root. Now, since the exponent of a is even (6), we can also remove a³ as a perfect square because (a³)² = a⁶.
Therefore, we can simplify the square root expression as follows:
√108a⁶ = √(2² × 3³ × a⁶) = √(4 × 9 × a⁶) = √4 × √9 × √a⁶ = 2 × 3 × a³ = 6a³.
This simplifies the original expression without perfect squares under the square root, giving us a simplified radical form.