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.