Question

How many roots does the following equation possess $3^{|x|(|2-|x||)} = 1$?

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

Answer: (B)

3

Answer 2

The given equation is $3^{|x|(|2-|x||)} = 1$.

For any base $a > 0$ and $a \neq 1$, if $a^b = 1$, then the exponent $b$ must be equal to 0. In this case, the base is 3, which is positive and not equal to 1. Therefore, the exponent must be 0:

$|x|(|2-|x||) = 0$

This equation holds true if either of the factors is zero:

1. $|x| = 0$
2. $|2-|x|| = 0$

Let's solve each case:

Case 1: $|x| = 0$

If the absolute value of $x$ is 0, then $x$ must be 0. So, $x = 0$ is one root.

Case 2: $|2-|x|| = 0$

If the absolute value of an expression is 0, then the expression itself must be 0. So, $2-|x| = 0$

Rearranging the term, we get:

$|x| = 2$

If the absolute value of $x$ is 2, then $x$ can be 2 or -2. So, $x = 2$ and $x = -2$ are two more roots.

Combining the roots from both cases, the distinct roots of the equation are $x = 0$, $x = 2$, and $x = -2$.

Thus, there are 3 roots for the given equation.