Question

How do you simplify $ {27^{ - \left( {\dfrac{2}{3}} \right)}} $ ?

Answer 1

Hint: In this question we have to simplify for a number which is given in the exponents form. To simplify the given number we have to use basic laws of exponents. And we also know both the square root and cube root of numbers or we have to know how to find the square root or cube root of a number.

Complete step-by-step answer:
Let us try to simplify the given number which is in exponent form. To simplify this we will need some basic laws of exponents. Here are the some basic laws of exponents are:
 $ {a^m} = (a \times a \times a \times ... \times a \times a) $ $ m $ $ times $ ( where $ a $ is base and $ m $ is exponent)
 $ {a^{ - m}} = \dfrac{1}{{{a^m}}} $
 $ {a^{m \cdot n}} = {({a^m})^n} $
We first apply property 3 which says that $ {a^{m \cdot n}} = {({a^m})^n} $ . After applying property 3 we have,
 $ {27^{ - \left( {\dfrac{2}{3}} \right)}} = {({27^{ - \left( {\dfrac{1}{3}} \right)}})^2} $
Now, we apply property 2 which says that $ {a^{ - m}} = \dfrac{1}{{{a^m}}} $ . After applying property 2 we have,
  $ {27^{ - \left( {\dfrac{2}{3}} \right)}} = {\left( {{{27}^{ - \left( {\dfrac{1}{3}} \right)}}} \right)^2} = {\left( {\dfrac{1}{{{{27}^{\left( {\dfrac{1}{3}} \right)}}}}} \right)^2} $
As we cube root of $ {27^{\dfrac{1}{3}}} = 3 $ .So we have,
 $ {27^{\left( { - \dfrac{2}{3}} \right)}} = {\left( {\dfrac{1}{3}} \right)^2} $
Now, we apply property 1 which says that $ {a^m} = (a \times a \times a \times ... \times a \times a) $ $ m $ $ times $ .After applying property 1, we have
 $ {27^{\left( { - \dfrac{2}{3}} \right)}} = {\left( {\dfrac{1}{3}} \right)^2} = \dfrac{1}{9} $ (Because $ {\left( {\dfrac{1}{3}} \right)^2} = \dfrac{1}{9} $ )
Hence the value of $ {27^{ - \left( {\dfrac{2}{3}} \right)}} $ in number form is $ \dfrac{1}{9} $.
So, the correct answer is “ $ \dfrac{1}{9} $ ”.

Note: To solve this type of question we need to know the definition of exponents and the basic properties of exponents. Also to solve you have to learn square, cube, cube root and square root of a number. Exponents are usually used to write the very big numbers and small numbers. For example: distance between the moon and the earth, size of atoms. Exponents are very useful in performing mathematical operations and handling of numbers.