If x = 3 + √8, then the value of x³ + 1/x³ is

For two numbers a and b whose product ab is known, the sum of cubes follows directly from expanding a sum cubed: a3 + b3 = (a + b)3 - 3ab(a + b). This identity converts a hard-to-compute sum of cubes into an easy computation using only the sum a + b and the product ab.

Here we take a = x and b = 1/x, so ab = x . (1/x) = 1 always - this is what makes the identity so useful for reciprocal-pair expressions like x3 + 1/x3.

Applying this to the given value of x:

1. Rationalize the reciprocal: since x = 3 + √8, multiply numerator and denominator by the conjugate to get 1/x = (3 - √8)/[(3 + √8)(3 - √8)] = (3 - √8)/(9 - 8) = 3 - √8.

2. Add x and 1/x: x + 1/x = (3 + √8) + (3 - √8) = 6.

3. Confirm the product: x . (1/x) = (3 + √8)(3 - √8) = 9 - 8 = 1, so ab = 1 in the identity above.

4. Substitute into the identity: x3 + 1/x3 = (x + 1/x)3 - 3(x)(1/x)(x + 1/x) = 63 - 3(1)(6) = 216 - 18.

5. Simplify: 216 - 18 = 198.

Cross-check with decimals: √8 ≈ 2.828, so x ≈ 5.828 and 1/x ≈ 0.172; x3 ≈ 197.995 and 1/x3 ≈ 0.005, and their sum is ≈ 198.000, matching the exact result.

So x3 + 1/x3 = 198.