Question

1:2√12+1:3√27+1:4√48

Answer 1

1/2√12+1/3√27+1/4√48=1/2√4*3+1/3√9*3+1/4√16*3=2*1/2√3+3*1/3√3+4*1/4√3=√3+√3+√3=3√3

Answer 2

To simplify the expression, we need to find the square root of 12, cube root of 27, and fourth root of 48, and then add the fractions.

1. Square root of 12: √12 = √(4 × 3) = √4 × √3 = 2√3

2. Cube root of 27: ∛27 = 3

3. Fourth root of 48: 4√48 = 4√(16 × 3) = 4√16 × 4√3 = 2 × 4√3 = 8√3

Now the expression becomes:

1:2√3 + 1:3 + 1:8√3

To add these fractions, we need to find a common denominator, which is the least common multiple (LCM) of the denominators (2, 3, and 8). The LCM of 2, 3, and 8 is 24.

Now, we rewrite each fraction with the common denominator:

(1/2)√3 + (1/3) + (1/8)√3

To combine the fractions, we convert them to have the same radical term:

(1/2)√3 + (8/24) + (3/24)√3

Next, we sum the fractions:

(1/2)√3 + (8+3)/24√3

(1/2)√3 + 11/24√3

Now, since both terms have the same radical (√3), we can combine them:

(1/2 + 11/24)√3

To add the fractions inside the parentheses, we find a common denominator, which is 24:

(12/24 + 11/24)√3

(23/24)√3

Therefore, the simplified expression is:

(23/24)√3