Question

Find the value of \[{{\left( {{2}^{-2}} \right)}^{-2}}\times {{\left( {{4}^{-2}} \right)}^{-4}}\times {{\left( {{8}^{-2}} \right)}^{4}}\] .

Answer 1

Hint: In the question, we have an expression and we have to find its value. We know the formula that, \[{{({{x}^{n}})}^{m}}={{x}^{mn}}\] . Using this formula in the given expression, we get \[{{(2)}^{4}}\times {{(4)}^{8}}\times {{(8)}^{-8}}\] . We can write 4 as \[{{2}^{2}}\] and 8 as \[{{2}^{3}}\] . Replace 4 by \[{{2}^{2}}\] and 8 by \[{{2}^{3}}\] in the given expression. Now, the expression can be solved further.

Complete step-by-step answer:
We have the expression, \[{{\left( {{2}^{-2}} \right)}^{-2}}\times {{\left( {{4}^{-2}} \right)}^{-4}}\times {{\left( {{8}^{-2}} \right)}^{4}}\] . First of all, we have to simplify this expression and then find its value.
We know the formula, \[{{({{x}^{n}})}^{m}}={{x}^{mn}}\] .
Using this formula, simplifying the first term of the expression. That is,
\[{{({{2}^{-2}})}^{-2}}={{2}^{-2\times -2}}={{2}^{4}}\]……………..(1)
Using this formula, simplifying the second term of the expression. That is,
\[{{({{4}^{-2}})}^{-4}}={{4}^{-2.-4}}={{4}^{8}}\]……………..(2)
Using this formula, simplifying the third term of the expression. That is,
\[{{({{8}^{-2}})}^{4}}={{8}^{-2.4}}={{8}^{-8}}\]……………..(3)
Replacing 4 by \[{{2}^{2}}\] and 8 by \[{{2}^{3}}\] in equation (2) and equation (3) we get,
\[{{4}^{8}}={{({{2}^{2}})}^{8}}\]…………(4)
\[{{8}^{-8}}={{({{2}^{3}})}^{-8}}\]……………(5)
Now, using the formula \[{{({{x}^{n}})}^{m}}={{x}^{mn}}\] in equation (4), we get
\[{{({{2}^{2}})}^{8}}={{2}^{16}}\]…………….(6)
Now, using the formula \[{{({{x}^{n}})}^{m}}={{x}^{mn}}\] in equation (4), we get
\[{{({{2}^{3}})}^{-8}}={{2}^{-24}}\]……………..(7)
Using equation (1), equation (6) and equation (7), transform the given expression.
\[\begin{align}
  & {{\left( {{2}^{-2}} \right)}^{-2}}\times {{\left( {{4}^{-2}} \right)}^{-4}}\times {{\left( {{8}^{-2}} \right)}^{4}} \\
 & ={{2}^{4}}\times {{2}^{16}}\times {{2}^{-24}} \\
\end{align}\]
We know the formula, \[{{x}^{m}}\times {{x}^{n}}\times {{x}^{p}}={{x}^{m+n+p}}\] .
Using this formula, we can solve the expression \[{{2}^{4}}\times {{2}^{16}}\times {{2}^{-24}}\].
\[\begin{align}
  & {{2}^{4}}\times {{2}^{16}}\times {{2}^{-24}} \\
 & ={{2}^{4+16-24}} \\
 & ={{2}^{-4}} \\
\end{align}\]
So, the value of the given expression is \[{{2}^{-4}}\] .

Note: We can also solve this question in another way. First, we have to remove the negative sign from the exponents. If we have to remove the negative sign from the exponents then we have to take reciprocal of the given number. Now, doing so in our given expression and solving it.
\[\begin{align}
  & {{\left( {{2}^{-2}} \right)}^{-2}}\times {{\left( {{4}^{-2}} \right)}^{-4}}\times {{\left( {{8}^{-2}} \right)}^{4}} \\
 & ={{\left( \dfrac{1}{{{2}^{2}}} \right)}^{-2}}\times {{\left( \dfrac{1}{{{4}^{2}}} \right)}^{-4}}\times {{\left( \dfrac{1}{{{8}^{2}}} \right)}^{4}} \\
 & ={{({{2}^{2}})}^{2}}\times {{({{4}^{2}})}^{4}}\times {{\left( \dfrac{1}{{{8}^{2}}} \right)}^{4}} \\
\end{align}\]
Now, using the formula \[{{({{x}^{n}})}^{m}}={{x}^{mn}}\] , we can simplify it.
\[\begin{align}
  & ={{({{2}^{2}})}^{2}}\times {{({{4}^{2}})}^{4}}\times {{\left( \dfrac{1}{{{8}^{2}}} \right)}^{4}} \\
 & ={{2}^{4}}\times {{4}^{8}}\times {{\left( \dfrac{1}{{{8}^{2}}} \right)}^{4}} \\
\end{align}\]
Replacing 4 by \[{{2}^{2}}\] and 8 by \[{{2}^{3}}\] in the above expression and solving it, we get
\[\begin{align}
  & ={{2}^{4}}\times {{4}^{8}}\times {{\left( \dfrac{1}{{{8}^{2}}} \right)}^{4}} \\
 & ={{2}^{4}}\times {{2}^{2.8}}\times {{\left( \dfrac{1}{{{2}^{3.2}}} \right)}^{4}} \\
 & ={{2}^{4}}\times {{2}^{16}}\times \left( \dfrac{1}{{{2}^{24}}} \right) \\
 & ={{2}^{20}}\times \left( \dfrac{1}{{{2}^{24}}} \right) \\
 & =\dfrac{1}{{{2}^{4}}} \\
 & ={{2}^{-4}} \\
\end{align}\]
Hence, the value of the given expression is \[{{2}^{-4}}\] .