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Algebra Level 2

Which of the equations below is/are false?

(A) : $\large\sqrt2 = 2^{1/2}$ .
(B) : $\large\sqrt{\sqrt2} = 2^{1/({2+2})}$ .
(C) : $\large\sqrt{\sqrt{\sqrt2}} = 2^{1/(2+2+2)}$ .

1 solution

Munem Shahriar
Jul 29, 2018

(A): $\sqrt 2 = 2^{\frac 12}$ is true.
(B): $\sqrt{\sqrt2} = 2^{\frac 14}$ is true. Proof: $\large \sqrt{\sqrt2} = (2^{\frac 12})^{\frac 12} = 2^{\frac 12 \times \frac 12} = 2^{\frac 14}.$
(C): $\sqrt{\sqrt{\sqrt 2}} = 2^{\frac 16}$ is false. Proof: $\large \sqrt{\sqrt{\sqrt 2}} = \{(2^{\frac 12})^{\frac 12}\}^{\frac 12} = 2^{\frac 12 \times \frac 12 \times \frac 12} = 2^{\frac 18} \ne 2^{\frac 16}.$

Only (C) is false.