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Difficulty: Very hardSurds and Indices

If x=7+43x = \sqrt{7 + 4\sqrt{3}}, y=7−43y = \sqrt{7 - 4\sqrt{3}}, and z=2+53+2−53z = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}, what is the value of the expression x3+y3x2+y2−z\frac{x^3 + y^3}{x^2 + y^2 - z}?

  1. 4Answer
  2. B
    5613\frac{56}{13}
  3. C
    133\frac{13}{3}
  4. D
    13

Answer

4
Expressing 7±437 \pm 4\sqrt{3} as perfect squares (2±3)2(2 \pm \sqrt{3})^2 simplifies xx to 2+32 + \sqrt{3} and yy to 2−32 - \sqrt{3}. This yields x+y=4x+y=4, xy=1xy=1, x2+y2=14x^2+y^2=14, and x3+y3=52x^3+y^3=52. For zz, using the cubic identity z3=a3+b3+3abzz^3 = a^3 + b^3 + 3ab z transforms the expression into z3+3z−4=0z^3 + 3z - 4 = 0, yielding the real solution z=1z = 1. Substituting these values gives 5214−1=4\frac{52}{14 - 1} = 4.

Step-by-Step Solution

1
Simplify the square root surds xx and yy
x=2+3x = 2 + \sqrt{3} and y=2−3y = 2 - \sqrt{3}
Since 7+43=4+3+2(2)(3)=(2+3)27 + 4\sqrt{3} = 4 + 3 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2, taking the square root gives 2+32 + \sqrt{3}. Similarly, 7−43=(2−3)27 - 4\sqrt{3} = (2 - \sqrt{3})^2.
2
Calculate fundamental algebraic combinations of xx and yy
x+y=4x + y = 4, xy=1xy = 1, x2+y2=14x^2 + y^2 = 14, and x3+y3=52x^3 + y^3 = 52
x+y=(2+3)+(2−3)=4x + y = (2+\sqrt{3}) + (2-\sqrt{3}) = 4. xy=(2+3)(2−3)=4−3=1xy = (2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1. x2+y2=(x+y)2−2xy=16−2=14x^2 + y^2 = (x+y)^2 - 2xy = 16 - 2 = 14. x3+y3=(x+y)3−3xy(x+y)=64−3(1)(4)=52x^3 + y^3 = (x+y)^3 - 3xy(x+y) = 64 - 3(1)(4) = 52.
3
Evaluate the nested cube root expression for zz
z=1z = 1
Let z=a+bz = a + b where a=2+53a = \sqrt[3]{2+\sqrt{5}} and b=2−53b = \sqrt[3]{2-\sqrt{5}}. Cubing both sides: z3=a3+b3+3ab(a+b)=(2+5)+(2−5)+3(2+5)(2−5)3z=4+34−53z=4−3zz^3 = a^3 + b^3 + 3ab(a+b) = (2+\sqrt{5}) + (2-\sqrt{5}) + 3\sqrt[3]{(2+\sqrt{5})(2-\sqrt{5})} z = 4 + 3\sqrt[3]{4-5} z = 4 - 3z. Solving z3+3z−4=0z^3 + 3z - 4 = 0 gives (z−1)(z2+z+4)=0(z-1)(z^2 + z + 4) = 0, whose unique real root is z=1z = 1.
4
Substitute all values into the given expression x3+y3x2+y2−z\frac{x^3 + y^3}{x^2 + y^2 - z}
\frac{52}{14 - 1} = \frac{52}{13} = 4
Direct numerical evaluation of the simplified algebraic components.

Key Concept

Simplification of Nested Square and Cube Surds using Algebraic Identities
Estimated Time:3m 0s
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Topics

Surds and Indices

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