If $$x=7+7^{\frac{2}{3}}+7^{\frac{1}{3}}$$​, then which of the options below is correct?

Correct option is C

Given: $$x = 7 + 7^{\frac{2}{3}} + 7^{\frac{1}{3}}$$

Formula Used:

(A + B)³= A³ + B³ + 3 AB (A + B)

(A - B)³= A³ - B³ - 3 AB (A - B)

a^m + aⁿ = a^m+n​

Solution:

x - 7 = $$7^{\frac{2}{3}} + 7^{\frac{1}{3}}$$

Taking Cube both side:

(x - 7)³= $$(7^{\frac{2}{3}} + 7^{\frac{1}{3}} )^3$$

x³ - 343 - 21x (x - 7) = 7² + 7 + 3 $$\times$$ $$7^{\frac{2}{3}}$$ $$\times$$ $$7^{\frac{1}{3}}$$ $$(7^{\frac{2}{3}} + 7^{\frac{1}{3}} )$$

x³- 21x² + 147x - 343 = 49 + 7 + 21 (x - 7)

x³- 21x² + 147x - 343 = 21x - 147 + 56

​x³- 21x² + 126 x - 252 = 0

Thus, the correct option is (c)