Question

How do you simplify \[{\left( {\dfrac{1}{{27}}} \right)^{\dfrac{{ - 2}}{3}}}\] ?

Answer 1

Hint: We write prime factorization of the number in the base and solve the term using the formula of exponents. Use the formula of exponents to cancel the terms within power so as to make the solution easier.
Formula used:
* Formula of exponents states that powers can be multiplied such that \[{\left( {{x^m}} \right)^{\dfrac{1}{n}}} = {x^{m \times \dfrac{1}{n}}} = {x^{\dfrac{m}{n}}}\]

* Prime factorization of a number is writing the number in multiples of its factors where all factors are prime numbers.
* When the base is same powers can be added i.e. \[{a^m} \times {a^n} = {a^{m + n}}\]

* If the power has negative sign, then we take reciprocal of the term i.e.
\[{a^{ - 1}} = \dfrac{1}{a}\]

Complete step-by-step answer:
We have to find the value of \[{\left( {\dfrac{1}{{27}}} \right)^{\dfrac{{ - 2}}{3}}}\]……. (1).
We can write this as \[{\left( {{{\left( {\dfrac{1}{{27}}} \right)}^{ - 1}}} \right)^{\dfrac{{ 2}}{3}}}\] (according to above said)
Since we know if the power has negative sign, then we take reciprocal of the term i.e.
\[{\left( {\dfrac{1}{{27}}} \right)^{ - 1}} = 27\]
\[ \Rightarrow {\left( {\dfrac{1}{{27}}} \right)^{\dfrac{{ - 2}}{3}}} = {\left( {27} \right)^{\dfrac{2}{3}}}\]
Substitute the value of
\[27 = 3 \times 3 \times 3 = {\left( 3 \right)^3}\] (prime factorization) in the \[{\left( {27} \right)^{\dfrac{2}{3}}}\]
We get \[ \Rightarrow {\left( {\dfrac{1}{{27}}} \right)^{\dfrac{{ - 2}}{3}}} = {\left( {{{\left( 3 \right)}^3}} \right)^{\dfrac{2}{3}}}\]
Use the property of exponents i.e.
\[{\left( {{x^m}} \right)^{\dfrac{1}{n}}} = {x^{m \times \dfrac{1}{n}}} = {x^{\dfrac{m}{n}}}\]
We write it as
\[ \Rightarrow {\left( {\dfrac{1}{{27}}} \right)^{\dfrac{{ - 2}}{3}}} = {\left( {{{\left( 3 \right)}^3}} \right)^{\dfrac{2}{3}}} = {3^{3 \times \dfrac{2}{3}}}\]
Cancel the same terms in numerator and denominator of the power we get
\[ \Rightarrow {3^{\dfrac{{3 \times 2}}{3}}} = {3^2}\]
\[ \Rightarrow \]\[{3^2} = 3 \times 3 = 9\]
Therefore the answer will be \[9\].
So, the correct answer is “ \[9\]”.

Note: Students might make mistake of solving the power that exists in the power as they find it difficult to break the term \[\dfrac{{ - 2}}{3}\].Whenever there is negative power, remember there can be chances of reciprocal of base. Keep in mind we should always write the term in the base in the smallest and simplest form after collecting its powers so we can cancel as many terms in the power as possible.