Question

If we have an expression as \[{{\left\{ {{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}} \right\}}^{?}}=256\], find the value of?

Answer 1

Hint: Assume the value of ? equal to x. Apply the formula given as: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], used in the topic ‘exponents and powers’, to simplify the expression inside the bracket in the left hand side. In the right hand side, write 256 as \[{{2}^{y}}\] using prime factorization of 256. Compare their bases and equate their exponents to find the value of x.

Complete step-by-step solution
Here, we have been provided with the expression: - \[{{\left\{ {{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}} \right\}}^{?}}=256\]. We have to find the value of the question mark.
Let us assume the value of the question mark (?) as x. So, the expression becomes \[{{\left\{ {{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}} \right\}}^{?}}={{\left\{ {{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}} \right\}}^{x}}\]. Now, we have to determine the value of x.
Considering left hand side of the expression, we have,
L.H.S = \[{{\left\{ {{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}} \right\}}^{x}}\]
Apply the identity of the topic ‘exponents and powers’ given as: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], where ‘a’ is called the base and ‘m’ and ‘n’ are exponents, we get,
\[\Rightarrow \] L.H.S = \[{{\left\{ {{2}^{4\times \dfrac{1}{2}}} \right\}}^{x}}\]
\[\Rightarrow \] L.H.S = \[{{\left\{ {{2}^{2}} \right\}}^{x}}\]
\[\Rightarrow \] L.H.S = \[{{2}^{2x}}\] - (1)
Now, in the right hand side, we have,
R.H.S = 256
Writing 256 as the product of its primes, we get,
\[\Rightarrow \] R.H.S = 256 = 2 \[\times \] 2 \[\times \] 2 \[\times \] 2 \[\times \] 2 \[\times \] 2 \[\times \] 2 \[\times \]2
\[\Rightarrow \] R.H.S = 256 = \[{{2}^{8}}\] - (2)
Equating equation (1) and (2), we get,
\[\Rightarrow {{2}^{2x}}={{2}^{8}}\]
Here, we can see that the base term on sides are equal, therefore removing the base 2 and comparing their exponents, we get,
\[\begin{align}
  & \Rightarrow 2x=8 \\
 & \Rightarrow x=4 \\
\end{align}\]
Hence, option (c) is the correct answer.

Note: One may note that we can also write both sides of the expression as a power of base 4 and equate \[{{4}^{x}}={{4}^{4}}\]. But it is always convenient to write the numbers as a product of their primes because sometimes we may get confused by not writing the numbers as their product of primes. We must remember some basic formulas of the topic exponent and power like: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\] and \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]. These formulas are used everywhere.