Question

How do you simplify ${729^{\dfrac{{ - 1}}{3}}}$?

Answer 1

Hint: The above question is based on the exponent expression.
Any variable or number when rose to some power then it is called exponent, be it positive or negative. It means how many times a number is multiplied by itself.
Using a few properties we will solve the given problem.

Complete step-by-step solution:
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)
$ \Rightarrow {({a^n})^m} = {a^{mn}}$ (When the base of the exponent has power on power then its power will be multiplied)
$ \Rightarrow {({a^n})^{\dfrac{1}{n}}} = a$ (When power on power is in the fraction, then if the powers are same they get cancelled)
Now, we will solve the given problem using the above written properties.
$ \Rightarrow {\left( {729} \right)^{ - \dfrac{1}{3}}}$ (Given exponent)
$ \Rightarrow \dfrac{1}{{{{\left( {729} \right)}^{\dfrac{1}{3}}}}}$ (Reciprocal of the given fraction)
$ \Rightarrow \dfrac{1}{{{{({{\left( 9 \right)}^3})}^{\dfrac{1}{3}}}}}$ (729 is the cube of 9)
$ \Rightarrow \dfrac{1}{9}$ (Powers were same, thus cancelled the two)

$\dfrac{1}{9}$ is the required answer.

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 ${10^{ - 10}}$.