Question

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

Answer 1

Hint: Assume the simplified form of the given expression as ‘E’. Write the base of ‘E’ as \[-{{3}^{3x}}=\left( -1\times 3\times {{x}^{3}} \right)\] and apply the formula of exponent and powers as: - \[{{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}\] and simplify the value. Now, use the formula \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] to simplify the value of \[{{\left( {{x}^{3}} \right)}^{-2}}\]. Finally, use the identity: - \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\] to get the answer.

Complete step by step answer:
Here, we have been provided with the expression \[{{\left( -{{3}^{3x}} \right)}^{-2}}\] and we are asked to simplify it.
Now, let us assume the value of the given expression as E. So, we have,
\[\Rightarrow E={{\left( -{{3}^{3x}} \right)}^{-2}}\]
We can write the base, i.e., \[\left( -{{3}^{3x}} \right)\] as \[\left( -1\times 3\times {{x}^{3}} \right)\], so we get,
\[\Rightarrow E={{\left( -1\times 3\times {{x}^{3}} \right)}^{-2}}\]
Breaking the terms we get, using the formula: - \[{{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}\],
\[\Rightarrow E={{\left( -1 \right)}^{-2}}\times {{\left( 3 \right)}^{-2}}\times {{\left( {{x}^{3}} \right)}^{-2}}\]
Now, applying the formula: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], we get,
\[\begin{align}
  & \Rightarrow E={{\left( -1 \right)}^{-2}}\times {{\left( 3 \right)}^{-2}}\times {{x}^{3\times \left( -2 \right)}} \\
 & \Rightarrow E={{\left( -1 \right)}^{-2}}\times {{\left( 3 \right)}^{-2}}\times {{x}^{-6}} \\
\end{align}\]
Using the conversion formula: - \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\], we get,
\[\begin{align}
  & \Rightarrow E=\dfrac{1}{{{\left( -1 \right)}^{2}}}\times \dfrac{1}{{{\left( 3 \right)}^{2}}}\times \dfrac{1}{{{x}^{6}}} \\
 & \Rightarrow E=\dfrac{1}{1}\times \dfrac{1}{9}\times \dfrac{1}{{{x}^{6}}} \\
 & \Rightarrow E=\dfrac{1}{9{{x}^{6}}} \\
\end{align}\]
Hence, the above expression represents the simplified form of the given exponential expression.

Note:
One may note that here we have used some basic formulas of the topic ‘exponents and powers’ to solve the question. You must remember some basic formulas such as: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}},{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}},{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] and \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\] because they are used everywhere. There is no easier method to solve the above question. Note that we can also write the simplified form as: - \[\dfrac{1}{9{{x}^{6}}}=\dfrac{x-6}{9}\]. Remember that: - \[{{\left( -1 \right)}^{n}}=1\] when ‘n’ is even and (-1) when ‘n’ is an odd integer.