Question

How to prove that 3^1/2*3^1/4*3^1/8 ... = 3 please help​

Answer 1

we have to prove that ,

$\quad3^{1/2}\times3^{1/4}\times3^{1/8}=3$

we know, $\boxed{\bf{x^m.x^n=x^{m+n}}}$

$\textbf{so,}LHS=3^{1/2}.3^{1/4}.3^{1/8}=3^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}}\\\\=3^{\frac{4}{8}+\frac{2}{8}+\frac{1}{8}}\\\\=3^{\frac{4+2+1}{8}}\\\\=3^{\frac{\cancel{8}}{\cancel{8}}}\\\\=3^1=3=RHS$

Answer 2

Answer:

3^(1/2 + 1/4 + 1/8 + ... ) = 3^1 = 3

Step-by-step explanation:

3^1/2 * 3^1/4 * 3^1/8 ... = 3

3^(1/2 + 1/4 + 1/8 + ... ) = 3

Now we look at 1/2 + 1/4 + 1/8 + ...

1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a geometric series

The sum of this series can be denoted in summation notation as:  1/2 + 1/4 + 1/8 + 1/16 + · · · = (1/2) / ( 1 - 1/2 ) = 1

-From here it is obvious that 3^(1/2 + 1/4 + 1/8 + ... ) = 3^1 = 3