Question

Which of the following number(s) is/are negative? Choose the most suitable option.

Answer 1

As we know, a negative base raised to an even power always gives a positive value. A negative base raised to an odd power always gives a negative value.

For $a<0,$

• $a^n$ is positive if $n$ is even.
• $a^n$ is negative if $n$ is odd.

Now, let's elaborate all the options.

$\bullet$ In $(–5{)}^2,$ the base $(–5)$ is negative and the exponent $\left(2\right)$ is even. Therefore, $(–5)^2$ has a positive value.

$\bullet$ In $(–7{)}^3,$ the base $(–7)$ is negative and the exponent $\left(3\right)$ is odd. Therefore, $(–7)^3$ has a negative value.

$\bullet$ $–6^8=–(6{)}^8$
This can also be written as $–6^8=–1\times (6{)}^8$.
A positive base raised to any power always gives a positive value. Therefore, $6^8$ is a positive number. However, a positive number when multiplied by a negative number generates a negative value. Therefore, $–6^8$ is a negative number.

$\therefore$ $(–5{)}^2$ and $9^3$ have positive values, and $(–7{)}^3$ and $–6^8$ have negative values.