Question

Find the value of ${{\left( {{27}^{-\dfrac{2}{3}}} \right)}^{\dfrac{1}{2}}}$?
(a) $\dfrac{1}{9}$
(b) $\dfrac{1}{3}$
(c) $\dfrac{1}{5}$
(d) None of these

Answer 1

Hint: Write the given base number 27 as the product of its prime factors with the help of prime factorization method. Now, write it in the exponential form if the factors are repeated. Use the formula of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify the expression. Finally, use the formula ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ to get the answer.

Complete step by step answer:
Here we have been provided with the expression ${{\left( {{27}^{-\dfrac{2}{3}}} \right)}^{\dfrac{1}{2}}}$ and we are asked to find its value. Here we will use some formulas of exponents and powers. Let us assume the given expression as ‘E’, so we have,
$\Rightarrow E={{\left( {{27}^{-\dfrac{2}{3}}} \right)}^{\dfrac{1}{2}}}$
Now, writing the base which is 27 as the product of its prime factors using the prime factorization method we get the expression as: -
\[\Rightarrow E={{\left( {{\left( 3\times 3\times 3 \right)}^{-\dfrac{2}{3}}} \right)}^{\dfrac{1}{2}}}\]
Since the factor 3 is repeated three times, so we can write it in the exponential form as: -
\[\Rightarrow E={{\left( {{\left( {{3}^{3}} \right)}^{-\dfrac{2}{3}}} \right)}^{\dfrac{1}{2}}}\]
Now, using the formula of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ we get,
\[\Rightarrow E={{\left( 3 \right)}^{3\times \left( -\dfrac{2}{3} \right)\times \dfrac{1}{2}}}\]
Cancelling the common factors in the exponents and simplifying we get,
\[\Rightarrow E={{\left( 3 \right)}^{-1}}\]
Using the formula of exponents given as ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ we get,
\[\therefore E=\dfrac{1}{3}\]

So, the correct answer is “Option b”.

Note: Here we have used some basic formulas of the topic ‘exponents and powers’ which must be remembered to solve the question. You must remember formulas like: - \[{{x}^{m}}\times {{x}^{n}}={{x}^{m+n}}\], \[{{x}^{m}}\div {{x}^{n}}={{x}^{m-n}}\], \[{{\left( {{x}^{m}} \right)}^{n}}={{x}^{m\times n}}\], \[{{x}^{-m}}=\dfrac{1}{{{x}^{m}}}\] etc, as they are used in certain other topics of mathematics. You must convert the base into the exponential form for which you must remember the prime factorization method otherwise it will be difficult to solve the question.