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Write the reciprocal of each of the following:
(i) ${{\left( 16 \right)}^{-7}}$
(ii) ${{\left( \dfrac{2}{3} \right)}^{-4}}$

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Last updated date: 21st Jul 2024
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Answer
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Hint: For solving this question you should know about the reciprocal of any number. The reciprocal of any term is defined as the negative one (-1) power of that whole term. That new term is defined as a reciprocal of that term. In this problem we will take the reciprocals of the given terms.

Complete step-by-step solution:
According to our question, it is asked of us to find the reciprocals of ${{\left( 16 \right)}^{-7}}$ and ${{\left( \dfrac{2}{3} \right)}^{-4}}$. As we know that the reciprocal of that value is equal to the inverse of that value or it is also said that it is equal to the -1 power of that whole term. And if we solve this then we get the final answers for the questions. So, if we look at our question, then:
(i) ${{\left( 16 \right)}^{-7}}$
Here our term is given as ${{\left( 16 \right)}^{-7}}$.
We can also write this as ${{\left( \dfrac{1}{16} \right)}^{7}}$ or it can be also said that,
${{\left( 16 \right)}^{-7}}={{\left( \dfrac{1}{16} \right)}^{7}}$
Now, according to the definition of reciprocal, we will find the reciprocal value of ${{\left( \dfrac{1}{16} \right)}^{7}}$ , which will be the same as the reciprocal value of ${{\left( 16 \right)}^{-7}}$.
So, reciprocal of ${{\left( 16 \right)}^{-7}}=\dfrac{1}{\left[ {{\left( 16 \right)}^{-7}} \right]}={{\left( 16 \right)}^{7}}$.
Or if we find the reciprocal of ${{\left( \dfrac{1}{16} \right)}^{7}}$, then $\dfrac{1}{{{\left( \dfrac{1}{16} \right)}^{7}}}={{16}^{7}}$.
So, both are equal.
(ii) ${{\left( \dfrac{2}{3} \right)}^{-4}}$
Here our term is given as ${{\left( \dfrac{2}{3} \right)}^{-4}}$ .
We can also write this as ${{\left( \dfrac{3}{2} \right)}^{4}}$ or it can be also said that,
${{\left( \dfrac{2}{3} \right)}^{-4}}={{\left( \dfrac{3}{2} \right)}^{4}}$
Now, according to the definition of reciprocal, we will find the reciprocal value of ${{\left( \dfrac{2}{3} \right)}^{-4}}$, which will be the same as the reciprocal value of ${{\left( \dfrac{3}{2} \right)}^{4}}$.
So, reciprocal of ${{\left( \dfrac{2}{3} \right)}^{-4}}=\dfrac{1}{{{\left( \dfrac{2}{3} \right)}^{-4}}}={{\left( \dfrac{2}{3} \right)}^{4}}$.
Or if we find the reciprocal of ${{\left( \dfrac{3}{2} \right)}^{4}}$, then $\dfrac{1}{{{\left( \dfrac{3}{2} \right)}^{4}}}={{\left( \dfrac{2}{3} \right)}^{4}}$.
So, both the values are equal.
So, the reciprocals are ${{\left( 16 \right)}^{7}}$ and ${{\left( \dfrac{2}{3} \right)}^{4}}$.

Note: While solving these types of questions you should be careful taking the reciprocal, because many times the division of that is not correct and many times the power gets the wrong sign and these minor mistakes make our questions wrong. So, always do the calculations properly.