Question

How do you solve for \[ - {625^{ - \dfrac{1}{4}}}\] ?

Answer 1

Hint:The given question have an fraction of power to the negative integer number, here the power is in fraction so to solve it either we can directly find the fourth root which is quite difficult sometime, or we can break the number into its factors and we can raise the power of number to its factors, and accordingly we can solve for it.

Complete step by step answer:
The given question is \[ - {625^{ - \dfrac{1}{4}}}\]. Now here first we have to find the factors of the given number which is six hundred twenty five, here the number is multiple of five so we will break the number into product of five, on solving we get:
\[- 625 = - \left( {5 \times 5 \times 5 \times 5} \right) \\
\Rightarrow - 625 = - ({5^4}) \\ \]
Here we got the number in the multiple of five, we see that when five is multiplied to four times then it gives the number six hundred twenty five, now we will put the factor value in the original question, and after solving we can get the required answer for the given question, on solving we get:
\[ - {625^{ - \dfrac{1}{4}}} = - {({5^4})^{ - \dfrac{1}{4}}}\]
Here we know that when the powers on a number is present one in bracket and another above bracket then both the power are multiplied to get the final result,
\[ - {625^{ - \dfrac{1}{4}}} = - {({5^4})^{ - \dfrac{1}{4}}} \\
\Rightarrow - {625^{ - \dfrac{1}{4}}} = - {(5)^{ - 4 \times \dfrac{1}{4}}} \\
\Rightarrow - {625^{ - \dfrac{1}{4}}} =- {(5)^{ - 1}} \\
\therefore - {625^{ - \dfrac{1}{4}}} = \dfrac{{ - 1}}{5}\]

Note:Here for this question it was easy to solve the number and to find the factors such that the power on factors can be solved with the given power in the question, otherwise direst solving for the fourth root is also an option but quite typical here.