#### Question

Find the value of \sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}

Question

Find the value of \sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}

Let

a =\sqrt[3]{7+5\sqrt{2} }

(1)

b = \sqrt[3]{7-5\sqrt{2} }

(2)

Addition of the equations after raising a and b to the 3rd power gives

a^3+b^3 = 14

(3)

multiplying (1) with (2) gives

ab = \sqrt[3]{7^2-(5\sqrt{2} )^2} =\sqrt[3]{-1} = -1

(4)

Transform the identity of perfect square of binomial to

(a+b)^3 = a^3+b^3+3ab(a+b)

(5)

Let

x = a+b

(6)

Substituting (3), (4) and (6) into (5) results in a depressed cubic equation

x^3+3x-14 = 0

(7)

By observation, x = 2 is one of the solution for equation (7)

Then, divide the polynomial x^3+3x-14 by x-2 to find another factor

(x-2)(x^2+2x+7) = 0

(8)

Since x^2+2x+7= (x+1)^2 + 6>0 , there is no real solution for the equation x^2+2x+7 = 0

Therefore, there's only one solution x = 2. Now we conclude

\sqrt[3]{7+5\sqrt{2} } +\sqrt[3]{7-5\sqrt{2} } =2

Transform the binomial with square root to the form of cube of binomial

7+5\sqrt{2}=(1+\sqrt{2})^3

7-5\sqrt{2}=(1-\sqrt{2})^3

Find cube root: \sqrt[3]{7+5\sqrt{2}}=1+\sqrt{2}

Find cube root: \sqrt[3]{7-5\sqrt{2}}=1-\sqrt{2}

Therefore

\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}=2