Final answer:
To compare the square roots of 180 and 102/8, convert the mixed number to an improper fraction. Simplify both square roots and compare. The comparison using > is correct.
Step-by-step explanation:
To compare the square roots of 180 and 102/8, we can convert the mixed number into an improper fraction. 102/8 can be written as 102 ÷ 8 = 12 remainder 6, so 102/8 = 12 + 6/8 = 12 ¾. Now, we can compare the two square roots: √180 and √12 ¾.
Simplifying both square roots, we have:
√180 = √(36 x 5) = √36 x √5 = 6√5
√12 ¾ = √(9 x 1 + ¾) = √9 x √(1 + ¾) = 3 x √(1 + ¾) = 3 x √(1 + 2/2) = 3 x √(3/2) = 3√(3/2)
Comparing the two square roots, we have 6√5 > 3√(3/2). Therefore, the comparison using > is correct.