Question

How do you simplify \[{n^{ - 5}}\] by using only positive exponents?

Answer 1

Hint: We need to apply the basic rules which were taught to us while we were learning about powers. After that we need to change the negative exponent here to the positive exponent so that it gets easy for us to solve.

Complete step by step solution:
From the question, we know that it is a negative exponent. To solve this question, we need to first change this negative exponent to a positive exponent. First, we should multiply the negative exponent with \[1\]. Multiplying the negative exponent \[{n^{ - 5}}\] will not change anything, but it makes it easier to see what is happening later on in this problem:
\[{n^{ - 5}} \times 1 = {n^{ - 5}}\]

Now, to solve this, we will change the negative exponent into positive exponent. We know that a negative exponent in the numerator is a positive exponent in the denominator. So, we can rewrite the \[{n^{ - 5}}\] in a positive exponent as well. According to the rules of powers we know that:
\[{(a)^{ - n}} = \dfrac{1}{{{{(a)}^n}}}\]

Now, according to this formula, we need to write \[{n^{ - 5}}\] in the denominator and make it a positive exponent which looks like:
\[{n^{ - 5}} = \dfrac{1}{{{n^5}}}\]
The \[1\] which we had multiplied earlier is now the numerator and \[{n^5}\] is the denominator. Now, we just need to simplify this. We will write \[n\] multiplied on itself \[5\] times, and then we will get:
\[ \therefore{n^{ - 5}} = \dfrac{1}{{n \times n \times n \times n \times n}}\]

Note: We should always remember that a negative exponent in the numerator is a positive exponent in the denominator. Here in this question, if \[n\] was any number, then it would have been multiplied on itself \[5\] times.We can say that an exponent refers to the number of times a number is multiplied or duplicated by itself. There are two types of exponents. They are positive and negative exponents.