Question

The base of an exponent is negative and the corresponding power is an odd number, which is a negative integer. What is the sign of the final exponent $?$

• A. Positive
• B. Negative
• C. Cannot be determined

Answer 1

The correct option is B Negative

$\underset{–}{\text{Need to find:}}$ Sign of the exponent with negative base and power with an odd number, which is a negative integer

As an example, let's find the value and sign of the exponent $(-3{)}^{-5}.$

In $(-3{)}^{-5}$, the base , i.e., $-3$ is negative, and the exponent, i.e., $-5$ is an odd number, which is a negative integer.

$(-3{)}^{-5}$ can be written as:
$(-3{)}^{-5}=\frac{1}{(-3{)}^5}(\because a^{-n}=\frac{1}{a^n})$

Now, we need to find the value of $(-3{)}^5$.

$(-3{)}^5$

$=\underset{–}{(-3)\times (-3)}\times \underset{–}{(-3)\times (-3)}\times (-3)$

$=\underset{–}{9\times 9}\times (-3)$

$=81\times (-3)$

$\Rightarrow (-3{)}^5=-243$

$\therefore \underset{–}{(-3{)}^{-5}=\frac{1}{(-3{)}^5}=\frac{1}{-243}=-\frac{1}{243}}$

Hence, the sign of the exponent with negative base and odd power, which is also a negative integer, is always negative.

$\therefore$ Option (b.) is the correct answer.

$\underset{–}{\text{Note:}}$
If an exponent has negative base and odd power, the sign of the exponent is always negative irrespective of the positive or negative integer in the odd power.