Exponentiation by Successive Square Roots

Algebra Level 3

$\large \begin{aligned} A = \sqrt{3} \\ B = \sqrt{A} \\ C = \sqrt{B} \end{aligned}$

Which expression is equivalent to $3^{0.875}$ ?

A^3 B^2 C

C^6

C

ABC

1 solution

Michael Huang
Feb 4, 2017

Observe that $0.875 = \dfrac{7}{8}$ , which can be expressed as $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8}$ , and that $B = \sqrt{A} = \sqrt{\sqrt{3}} = 3^{1/4}$ and $C = \sqrt{B} = \sqrt{\sqrt{\sqrt{3}}} = 3^{1/8}$ . Multiplication of common bases shows that $\begin{array}{rl} 3^{0.875} &= 3^{1/2 + 1/4 + 1/8}\\ &= 3^{1/2}3^{1/4}3^{1/8}\\ &= \boxed{ABC} \end{array}$

Exponentiation by Successive Square Roots

1 solution

0 pending reports