Question

How do you simplify \[{{\left( \dfrac{81}{16} \right)}^{-\dfrac{1}{4}}}\] ?

Answer 1

Hint: To simplify \[{{\left( \dfrac{81}{16} \right)}^{-\dfrac{1}{4}}}\] , we need to solve the given form step by step, initially starting with removing the negative sign by writing the reciprocal of the given fraction. Now we have to apply the power given to the numerator and denominator separately. By solving the power by expanding the terms in numerator and denominator with their factors. We get the required simplified form of the given problem.

Complete step by step solution:
The given problem for simplification is as follows \[{{\left( \dfrac{81}{16} \right)}^{-\dfrac{1}{4}}}\]
We know the identity of exponents, \[{{\left( \dfrac{a}{b} \right)}^{-x}}\] = \[{{\left( \dfrac{b}{a} \right)}^{x}}\]
So, \[{{\left( \dfrac{81}{16} \right)}^{-\dfrac{1}{4}}}\]
\[\Rightarrow \]\[{{\left( \dfrac{16}{81} \right)}^{\dfrac{1}{4}}}\]
Since, from the identity of exponentials we have, \[{{\left( \dfrac{a}{b} \right)}^{m}}\]= \[\left( \dfrac{{{a}^{m}}}{{{b}^{m}}} \right)\]
\[\Rightarrow \] \[\left( \dfrac{{{16}^{\dfrac{1}{4}}}}{{{81}^{\dfrac{1}{4}}}} \right)\]
\[\Rightarrow \]\[\left( \dfrac{\sqrt[4]{16}}{\sqrt[4]{81}} \right)\]
We also know that \[{{a}^{\dfrac{1}{m}}}\] = \[\sqrt[m]{a}\]
The factorisation of the numbers 16 and 81 should be made as follows,
16 = 2 \[\times \] 2 \[\times \] 2 \[\times \] 2 and 81 = 3 \[\times \] 3 \[\times \] 3 \[\times \] 3
\[\Rightarrow \]\[\left( \dfrac{\sqrt[4]{2\times 2\times 2\times 2}}{\sqrt[4]{3\times 3\times 3\times 3}} \right)\]
\[\Rightarrow \]\[\dfrac{2}{3}\]
Hence, when the above problem \[{{\left( \dfrac{81}{16} \right)}^{-\dfrac{1}{4}}}\] is simplified we get to the value of \[\dfrac{2}{3}\].

Note: It seems very simple when you go through the solution but a small mistake in using the law of exponents may result in a wrong answer. It requires the skill to be well versed in finding the factors of the numbers by basic factorisation methods of the numbers like 16 and 81. Or you should know to find the fourth root of the numbers directly, if you cannot find the factors of the numbers. This problem can also be solved without taking the reciprocal initially. We can solve the fourth root first and then we can solve the negative power present. Even this method will yield us the same answer as above.