How do you divide and simplify this?

squareroot of 162a/6a^3

Thanks

sqrt[162a/(6a^3)] = sqrt (27/a^2) = (sqrt 27)/a

To divide and simplify the expression sqrt(162a/6a^3), follow these steps:

Step 1: Simplify the numerator and the denominator separately.

In the numerator, simplify the square root of 162a. Since 162 is not a perfect square, you can break it down into its prime factors:
162 = 2 * 3 * 3 * 3 * 3 = 2 * (3^4)

The square root of 162 can then be simplified as:
sqrt(162) = sqrt[2 * (3^4)] = sqrt(2) * (3^2) = 3 * sqrt(2)

In the denominator, you have 6a^3, which is already simplified.

Step 2: Divide the simplified numerator by the denominator.

sqrt(162a/6a^3) = (3 * sqrt(2))/(6a^3)

Step 3: Further simplify the expression.

To simplify further, divide both the numerator and denominator by the common factor, 3:

(3 * sqrt(2))/(6a^3) = sqrt(2)/(2a^3)

Final Answer: The simplified expression is sqrt(2)/(2a^3).