Question

How do you rewrite $\dfrac{1}{{{8^6}}}$ with negative exponents?

Answer 1

Hint: The above problem has a reciprocal of fraction with positive exponent, which will become negative by taking reciprocal of the denominator.
When the power of the exponent is positive and the base is shifted from denominator to numerator sign of the power changes.
Using the above concept we will solve the problem.

Complete step-by-step solution:
Let’s explain a few more concepts regarding the concept of exponents.
Generally, when an integer or any variable is raised to some power it is called an exponent, which means that the number of times a number is multiplied by itself.
We will discuss a few methods to solve any problem having exponents.
$ \Rightarrow {a^{ - m}} = \dfrac{1}{{{a^m}}}$ , a is the base here and m is the exponent. (When an exponent is in negative power it will become positive when it’s reciprocal is taken)
$ \Rightarrow {a^m} \times {a^n} = {a^{m + n}}$ (When any exponent having different values is written with the same base then the exponents are added).
$ \Rightarrow {a^0} = 1$ (Any variable or integer having zero as its exponent will give 1 as its value always)
Now, we will make the exponent negative by using the above properties.
$ \Rightarrow \dfrac{1}{{{8^6}}}$ (Given fraction)
$ \Rightarrow \dfrac{1}{{\dfrac{1}{{{8^6}}}}}$ (We have taken reciprocal of the given fraction)
$ \Rightarrow {8^{ - 6}}$ (Exponent with negative power)

$\dfrac{1}{{{8^6}}}$ with negative exponent is equal to ${8^{ - 6}}$.

Note: Exponential powers are not only used in mathematical calculations but are generally used to represent very large decimal values or values having large numbers of zeros in it such as distance of earth from the sun ($15 \times {10^{10}}$m, representation is easy to write), similarly in chemistry radius of atoms are very small which are generally measure Armstrong with ${10^{ - 10}}$.