The number (27+200^1/2)^1/2 can be simplified to the form a+b√, where a and b are positive integers. Find the product ab.

To simplify the given expression, we can follow these steps:

Step 1: Evaluate the innermost expression first, which is the square root of 200.

√200 = √(100 * 2) = √(100) * √(2) = 10√2

Step 2: Substitute the simplified value back into the original expression.

(27 + 10√2)^(1/2)

Step 3: Further simplify the expression by multiplying the terms inside the parentheses by the square root exponent.

(27^(1/2) + (10√2)^(1/2))

Step 4: Evaluate the square root of 27 and simplify the square root of 10.

√27 = √(9 * 3) = √(9) * √(3) = 3√3

√10 = √(2 * 5) = √(2) * √(5) = √(10)

Step 5: Substitute the simplified values back into the expression.

(3√3 + √10)

Step 6: To simplify further, we need to rationalize the denominator by multiplying the expression by its conjugate.

(3√3 + √10) * (3√3 - √10) / (3√3 - √10)

Using the difference of squares formula, (a^2 - b^2) = (a + b)(a - b):

((3√3)^2 - (√10)^2) / (3√3 - √10)

(9 * 3 - 10) / (3√3 - √10)

(27 - 10) / (3√3 - √10)

17 / (3√3 - √10)

Step 7: To rationalize the denominator, we can multiply the top and bottom of the fraction by the conjugate of the denominator.

[(17 / (3√3 - √10)) * ((3√3 + √10) / (3√3 + √10))]

(17 * (3√3 + √10)) / ((3√3 - √10) * (3√3 + √10))

(51√3 + 17√10) / (9 * 3 - (√10)^2)

(51√3 + 17√10) / (27 - 10)

(51√3 + 17√10) / 17

Step 8: Simplify the expression by factoring out the common factor of 17 from the numerator.

(17(3√3 + √10)) / 17

The 17 in the numerator and the denominator cancel each other out, leaving us with:

3√3 + √10

Now we can determine the product ab, where a and b are the coefficients of the radical terms:

a = 3
b = 1

ab = 3 * 1 = 3

Therefore, the product ab is equal to 3.