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    Which of the following is INCORRECT?
    Question

    Which of the following is INCORRECT?

    A.

    312+312+312=312×312×3123^\frac{1}{2}+3^\frac{1}{2}+ 3^\frac{1}{2}= 3^\frac{1}{2}× 3^\frac{1}{2}× 3^\frac{1}{2}

    B.

    413+413+413+413=413×413×413×413 4^\frac {1}{3}+4^\frac {1}{3}+4^\frac {1}{3}+4^\frac {1}{3}= 4^\frac {1}{3}× 4^\frac {1}{3}× 4^\frac {1}{3}× 4^\frac {1}{3} ​​

    C.

    514+514+514+514+514=514×514×514×514×5145^\frac {1}{4}+5^\frac {1}{4}+5^\frac {1}{4}+5^\frac {1}{4}+5^\frac {1}{4}= 5^\frac {1}{4}× 5^\frac {1}{4}× 5^\frac {1}{4}× 5^\frac {1}{4}× 5^\frac {1}{4}​​

    D.

    616+616+616+616+616+616=616×616×616×616×616×6166^\frac{1}{6}+ 6^\frac{1}{6}+6^\frac{1}{6}+6^\frac{1}{6}+6^\frac{1}{6}+6^\frac{1}{6}= 6^\frac{1}{6}× 6^\frac{1}{6}×6^\frac{1}{6}×6^\frac{1}{6}×6^\frac{1}{6}×6^\frac{1}{6}​​

    Correct option is D

    Solution:-

    312+312+312=312×312×3123^\frac{1}{2}+3^\frac{1}{2}+ 3^\frac{1}{2}= 3^\frac{1}{2}× 3^\frac{1}{2}× 3^\frac{1}{2}

    Left Side = 31/2+31/2+31/2=331/2=31/2+1=33/23^{1/2} + 3^{1/2} + 3^{1/2} = 3 \cdot 3^{1/2} = 3^{1/2 + 1} = 3^{3/2}

    Right side = 31/2×31/2×31/2=3(1/2)+(1/2)+(1/2)=33/23^{1/2} \times 3^{1/2} \times 3^{1/2} = 3^{(1/2) + (1/2) + (1/2)} = 3^{3/2} correct

    ​​413+413+413+413=413×413×413×4134^\frac {1}{3}+4^\frac {1}{3}+4^\frac {1}{3}+4^\frac {1}{3}= 4^\frac {1}{3}× 4^\frac {1}{3}× 4^\frac {1}{3}× 4^\frac {1}{3}

    Right side = (51/4)5=55/4\left(5^{1/4}\right) 5 = 5^{5/4} correct

    Left Side =66(6)=6(6+(6))=6(76)6 \cdot 6 - (-6) = 6 \left( 6 + (-6) \right) = 6 \left( \frac{7}{6} \right) 

    ​Right side = (6(66))=6\left( 6 \left( \frac{6}{6} \right) \right) = 6   Incorrect

    Thus, the correct answer is (D)

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