Question

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

Answer 1

Hint:
For solving these types of questions it is important that we know the laws of exponents and should properly know how to use them.

Formula used :
1) \[{{\left( \dfrac{x}{y} \right)}^{-1}}=\dfrac{y}{x}\]
2) \[{{\left( \dfrac{x}{y} \right)}^{m}}=\dfrac{{{x}^{m}}}{{{y}^{m}}}\] and
3) \[{{\left( x \right)}^{mn}}={{\left( {{\left( x \right)}^{m}} \right)}^{n}}\]

Complete step by step solution:
Let the given expression be \[E={{\left( -\dfrac{2}{3} \right)}^{-3}}\]. We are asked to simplify the given expression.
We know that, \[{{\left( x \right)}^{mn}}={{\left( {{\left( x \right)}^{m}} \right)}^{n}}\] . So using this law of exponent, we can rewrite the given expression E as:
\[E={{\left( -\dfrac{2}{3} \right)}^{-3}}={{\left( {{\left( -\dfrac{2}{3} \right)}^{3}} \right)}^{-1}}\]
We also know that, \[{{\left( \dfrac{x}{y} \right)}^{m}}=\dfrac{{{x}^{m}}}{{{y}^{m}}}\] . So using this law of exponent, we can rewrite the given expression E as:
\[E={{\left( {{\left( -\dfrac{2}{3} \right)}^{3}} \right)}^{-1}}={{\left( \dfrac{{{\left( -2 \right)}^{3}}}{{{3}^{3}}} \right)}^{-1}}={{\left( \dfrac{-8}{27} \right)}^{-1}}\]
We also know that, \[{{\left( \dfrac{x}{y} \right)}^{-1}}=\dfrac{y}{x}\] . So using this law of exponent, we can rewrite the given expression E as:
\[E={{\left( \dfrac{-8}{27} \right)}^{-1}}=\left( \dfrac{27}{-8} \right)=\left( -\dfrac{27}{8} \right)\]

Therefore, \[{{\left( -\dfrac{2}{3} \right)}^{-3}}=\left( -\dfrac{27}{8} \right)\] is the final answer.

Note:
While solving such questions, one should have a good knowledge about the Laws of Exponents. Also, one should be very keen while applying or using these properties/laws. At the end, one should check if the numerator and the denominator have any common factors or not, because if they do, then they should be cancelled as the answer has to be reported in its simplest form.