Find it.

Algebra Level 2

What is the value of the given expression

26 15 3 3 + 26 + 15 3 3 ? \sqrt[3]{26-15\sqrt{3}} + \sqrt[3]{26+15\sqrt{3}} ?


The answer is 4.

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3 solutions

Let the entire expression be equal to x x .Then we have that: x 3 = 26 + 15 3 + 26 15 3 + 3 ( 26 15 3 3 ) ( 26 + 15 3 3 ) ( 26 15 3 3 + 26 + 15 3 3 ) x^3=26+15\sqrt{3}+26-15\sqrt{3}+3\left(\sqrt[3]{26-15\sqrt{3}}\right)\left(\sqrt[3]{26+15\sqrt{3}}\right)\left(\sqrt[3]{26-15\sqrt{3}}+\sqrt[3]{26+15\sqrt{3}}\right) Since 26 15 3 3 + 26 + 15 3 3 = x \sqrt[3]{26-15\sqrt{3}}+\sqrt[3]{26+15\sqrt{3}}=x ,so: x 3 = 52 + 3 × 676 675 3 × x x 3 = 52 + 3 1 3 x x 3 = 52 + 3 x x 3 3 x 52 = 0 x^3=52+3\times\sqrt[3]{676-675}\times x\\ x^3=52+3\sqrt[3]{1}x\\x^3=52+3x\\ x^3-3x-52=0 So we have reduced the question to finding a root of the equation.In this case, the Rational Root Test reveals that 4 4 is a root.However this solution is incomplete if we stop here and don't examine the other 2 roots.Dividing x 3 3 x 52 x^3-3x-52 by x 4 x-4 ,we get x 2 + 4 x + 13 x^2+4x+13 .Upon checking the discriminant: b 2 4 a c = 4 2 4 ( 1 ) ( 13 ) = 16 52 = 36 b^2-4ac=4^2-4(1)(13)=16-52=-36 As the discriminant is smaller than 0,so the quadratic equation has non-real complex roots,but the expression in the question clearly is a real number.Hence we are left with 1 solution,that is 4 \boxed{4}

For those who find the firstmost equation difficult to grasp,remember that ( a + b ) 3 = a 3 + b 3 + 3 a b ( a + b ) (a+b)^3=a^3+b^3+3ab(a+b)

Abdur Rehman Zahid - 6 years, 3 months ago
Aman Gabba
Dec 3, 2015

(26-15 * 3^0.5) = (2-3^0.5)^3; (26-15 * 3^0.5)^(1/3) = (2-3^0.5); 2-(3^0.5)+2+(3^0.5) = 4

26 15 3 = ( 2 3 ) 3 26 - 15\sqrt{3}=(2-\sqrt{3})^3

26 + 15 3 = ( 2 + 3 ) 3 26 + 15\sqrt{3}=(2+\sqrt{3})^3

26 15 3 3 + 26 + 15 3 3 = ( 2 3 ) + ( 2 + 3 ) = 4 \sqrt[3]{26 - 15\sqrt{3}} + \sqrt[3]{26 + 15\sqrt{3}}=(2-\sqrt{3})+(2+\sqrt{3})=4

Aman Gabba - 5 years, 6 months ago
Gagan Raj
Feb 17, 2015

The first part of the question is 0.

The second part of the question is 3.713.

The answer is 0 + 3.713 = 3.713

But the answer does not support decimals.

Thus the answer is rounded off to the nearest integer.

So the answer is 4.

@Gagan Raj According to me,this is not a proper solution.

Abdur Rehman Zahid - 6 years, 3 months ago

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