If x=7-4√3 thenx^3+1/x^3=(7-4√3)^3+1/(7-4√3)^3 =2072

Calculation of x^3:

To find the value of x^3, we need to cube the given value of x, which is x = 7 - 4√3.

We can calculate x^3 using the formula (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. In this case, a = 7 and b = 4√3.

Step 1: Calculate a^3
a^3 = 7^3 = 343

Step 2: Calculate 3a^2b
3a^2b = 3 * (7)^2 * (4√3) = 3 * 49 * 4√3 = 588√3

Step 3: Calculate 3ab^2
3ab^2 = 3 * 7 * (4√3)^2 = 3 * 7 * 4 * 3 = 252

Step 4: Calculate b^3
b^3 = (4√3)^3 = 4^3 * (√3)^3 = 64 * 3√3 = 192√3

Step 5: Substitute the values into the formula
x^3 = a^3 - 3a^2b + 3ab^2 - b^3
= 343 - 588√3 + 252 - 192√3

Simplifying the expression, we can combine the like terms:
x^3 = (343 + 252) - (588√3 + 192√3)
= 595 - 780√3

Therefore, the value of x^3 is 595 - 780√3.

Calculation of 1/x^3:

To find the value of 1/x^3, we need to take the reciprocal of x^3.

Reciprocal of a number x is 1/x.

Step 1: Calculate the reciprocal of x^3
1/x^3 = 1/(595 - 780√3)

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is (595 + 780√3).

Step 2: Multiply the numerator and denominator by the conjugate of the denominator
1/x^3 = (1/(595 - 780√3)) * ((595 + 780√3)/(595 + 780√3))

Expanding the numerator and denominator, we get:
1/x^3 = (595 + 780√3)/(595^2 - (780√3)^2)

Simplifying further, we can calculate the values:
1/x^3 = (595 + 780√3)/(595^2 - 780^2 * 3)
= (595 + 780√3)/(354025 - 1825200)
= (595 + 780√3)/(-1471175)

Therefore, the value of 1/x^3 is (595 + 780√3)/(-1471175).