The value of 1 / √(15 - 6√6) is closest to:

CONCEPT

For a surd in a square root, first see whether the radicand can be written as a perfect-square form such as (a - b)2 or (√m - √n)2. Then simplify the square root before taking the reciprocal.

When the expression asks for the closest option, compute the simplified decimal only after the exact radical form has been reduced.

APPLICATION

1. Write the denominator as √(15 - 6√6). We look for numbers m and n such that m + n = 15 and 2√(mn) = 6√6.

2. From 2√(mn) = 6√6, we get √(mn) = 3√6, so mn = 54. The pair m = 9 and n = 6 satisfies m + n = 15 and mn = 54.

3. Therefore 15 - 6√6 = (√9 - √6)2 = (3 - √6)2.

4. So √(15 - 6√6) = 3 - √6, because 3 is greater than √6 and the denominator is positive.

5. Now rationalize: 1 / (3 - √6) = (3 + √6) / (9 - 6) = (3 + √6) / 3.

6. Using √6 ≈ 2.449, the value is (3 + 2.449) / 3 ≈ 5.449 / 3 ≈ 1.816.

CROSS-CHECK / CONTRAST

• 2.4 is too high because it is about 0.58 above the computed value.

• 1.2 is too low because it is about 0.62 below the computed value.

• 1 is farther below the computed value than the nearest decimal option.

• The closest listed value to about 1.816 is 1.8.

Result: the value of the expression is closest to 1.8.