If 3a + 2b = 27 and 27a³ + 8b³ = 1458, then find 2ab.

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Given:
3a + 2b = 27  ...(1)
27a³ + 8b³ = 1458 ...(2)

Notice that 27a³ = (3a)³ and 8b³ = (2b)³.
So, equation (2) can be written as:
(3a)³ + (2b)³ = 1458

From equation (1), 3a + 2b = 27.

Recall the identity for sum of cubes:
(x + y)³ = x³ + y³ + 3xy(x + y)

Let x = 3a and y = 2b.
Then:
(3a + 2b)³ = (3a)³ + (2b)³ + 3*(3a)*(2b)*(3a + 2b)

Substitute values:
27³ = 1458 + 3*(3a)*(2b)*27

Calculate 27³:
27³ = 27 * 27 * 27 = 19683

So:
19683 = 1458 + 3 * 3a * 2b * 27

Simplify the right side:
3 * 3a * 2b * 27 = 3 * 3 * 2 * 27 * ab = 486 * ab

So:
19683 = 1458 + 486 * ab

Subtract 1458 from both sides:
19683 - 1458 = 486 * ab
18225 = 486 * ab

Divide both sides by 486:
ab = 18225 / 486

Calculate:
486 * 37 = 17982
486 * 38 = 18468
So ab is between 37 and 38.
Calculate exact:
18225 / 486 = 37.5

Therefore, ab = 37.5

We need to find 2ab:
2ab = 2 * 37.5 = 75

Answer: 75

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