Question

The value of ${( - 2)^7}$ is:
(A) $ + 64$
(B) $ - 64$
(C) $ - 128$
(D) $ - 14$

Answer 1

Hint: Here we are given a number that is raised to another number. This is called exponentials and here the base value is ‘$ - 2$’ and the power or value of exponential here is ‘$7$’. Also depending on the power (if it’s even or odd) we can find out the sign that will be assigned to the final answer easily.

Complete step by step solution:
There are a number of ways we can solve this problem but let us proceed with a systematic method.
The given question represents an exponential term: ${( - 2)^7}$
Every exponential term has two parts, the base and the power or exponential;
Here the base $ = - 2$
And the power $ = 7$
This means that the number $2$ has to be multiplied by itself $7$ times.
Let us proceed to solve the exponential ${( - 2)^7}$;
We can ignore the sign of the base value for now and come back to it later.
Let us first multiply $2$ by itself $7$ times, for this we split the exponential in this way;
$ \Rightarrow {(2)^7} = {(2)^3} \times {(2)^3} \times {(2)^1}$
Since by law of exponentials: ${(a)^b} \times {(a)^c} = {(a)^{b + c}}$
We split it in this way to be able to multiply the terms easily in our mind or using mental Maths, since we have already memorized the value of ${2^3} = 8$.
Simplifying will look like this;
$ \Rightarrow {(2)^7} = 8 \times 8 \times 2$
Then the value will be:
$ \Rightarrow {(2)^7} = 128$
But we were given $ - 2$ as the base and we had kept aside the negative sign.
Now to predict the sign required we just need to look at the power in the exponential term.
If the power is an odd number then the ($ - $) sign of the base will be there in the final answer also.
If the power is an even number, the sign of the final result becomes positive ($ + $).
The power here was $7$ which is odd, so the final value $128$ will take a negative sign.
So the final answer becomes;
$ \Rightarrow {( - 2)^7} = - 128$
Let us look at the options:
Option (A) $ + 64$ is incorrect. This is because when we look at the power in the exponential term, we know that it is odd so the final answer has to contain a negative sign, but here sign is positive so we can reject this option.
Option (B) $ - 64$ is incorrect. Here the sign is negative which is correct but $64$ is the wrong answer since ${2^6} = 64$ so here the power is wrong, it was supposed to be $7$.
Option (C) $ - 128$ is correct. This is because the sign is correctly given and the value $ - 128$ is found to be the answer when the base $ - 2$ is raised to the power $7$.
Option (D) $ - 14$ is incorrect. This is because we get $ - 14$ when we take the product of $ - 2$ and $7$, not when the base $ - 2$ is raised to the power $7$.

Therefore, The value of ${( - 2)^7}$ is $-128$. So, the right option is option (C).

Note:
The above method is one way to solve the problem but there are other ways we can solve this question. For exponential questions if the power raised is small then we can just multiply the base number repeatedly by itself for the number of times mentioned in the power. Example if the question is ${( - 3)^2}$, it can also be written as ${( - 3)^2} = - 3 \times - 3$ so that the answer will be ${( - 3)^2} = 9$.