First slide

Theory of expressions

Question

The number of real solutions of the equation $27^{1 / x} + 12^{1 / x} = 2 (8^{1 / x})$ is

Moderate

A

0
B

1
C

infinite
D

none of these

Solution

We can write (1) as
${(\frac{27}{8})}^{1 / x} + {(\frac{12}{8})}^{1 / x} = 2$
or ${(\frac{3}{2})}^{3 / x} + {(\frac{3}{2})}^{1 / x} = 2$ (1)
As $3 / 2 > 1, (3 / 2)^{t} > 1$ if $t > 0$
and $(3 / 2)^{t} < 1$ if $t < 0$ .
Thus, ${(\frac{3}{2})}^{3 / x} + {(\frac{3}{2})}^{1 / x} < 2$ if $x > 0$ .
Therefore, (1) is possible if 1/x = 0. But this is not true for any real value of x.

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