To what power should (-2) be raised to get 64?

Question

Vedantu Content Team · Accepted Answer

Hint: In the above problem, we are asked to find the exponent of -2 in such a way so that after putting exponent on -2, the answer would be 64. To get the correct power of -2, we are going to use a hit and trial method. First we are going to take 1 as an exponent and then see what answer we are getting then we will take exponent as 2 then 3 then 4 till we will get the value as 64.Complete step by step answer:We know that power to any number means we are going to multiply that number by itself that many times.For e.g. let us take a number 3 and let’s put a power of 2 on it then we are going to multiply 3 two times.${{3}^{2}}=3\times 3$ Similarly, if the power is 3 then we are going to multiply 3 three times.${{3}^{3}}=3\times 3\times 3$ Now, in the above problem, we are going to find the power of -2 such that after applying that power, the answer should be 64.Let us take the power of -2 as 1. Applying the power of 1 on -2 we get,${{\left( -2 \right)}^{1}}=-2$ Single power means the number itself.Now, let us take the power as 2 on -2 we get,${{\left( -2 \right)}^{2}}=-2\times \left( -2 \right)$ We know that, when the odd number of minus signs are multiplied then their multiplication becomes positive.${{\left( -2 \right)}^{2}}=4$Taking power as 3 on -2 we get,${{\left( -2 \right)}^{3}}=-2\times \left( -2 \right)\times \left( -2 \right)$ We know that, when the minus signs are in odd numbers then their multiplication result is negative.${{\left( -2 \right)}^{3}}=-8$ Taking power as 4 on -2 we get,$\begin{align}  & {{\left( -2 \right)}^{4}}=-2\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right) \\  & \Rightarrow {{\left( -2 \right)}^{4}}=16 \\ \end{align}$Still, we are not getting 64 so now; we are taking power as 5 on -2 we get,$\begin{align}  & {{\left( -2 \right)}^{5}}=-2\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right) \\  & \Rightarrow {{\left( -2 \right)}^{5}}=-32 \\ \end{align}$Taking power as 6 on -2 we get,$\begin{align}  & {{\left( -2 \right)}^{5}}=-2\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right)\times \left( -2 \right) \\  & \therefore {{\left( -2 \right)}^{6}}=64 \\ \end{align}$Hence, the power is 6 raised this power onto -2 will give 64.Note:As you can see in the above solution that odd powers on -2 is giving a negative answer so you can skip taking odd powers in trial and rather take the even powers. By considering only even powers will save your time in the examination. It’s better to remember the exponents of 2 from 2 to 10. If you would have known that the exponent of 2 as 6 will give 64 then you can save time in examination.