Question

Express ${{16}^{-2}}$ as a power with base 4.

Answer 1

Hint: For solving this question firstly, we will see one of the basic formula of exponents that if ${{a}^{b}}=c$ then, ${{c}^{d}}={{\left( {{a}^{b}} \right)}^{d}}={{a}^{bd}}$ . After that, we will express the number 16 as a power with base 4. Then, we will be able to express ${{16}^{-2}}$ as a power with base 4 easily and give the correct answer for this question.

Complete step-by-step answer:
Given:
We have to express ${{16}^{-2}}$ as a power with base 4.
Now, before we proceed we should know that if ${{a}^{b}}=c$ . Then,
${{c}^{d}}={{\left( {{a}^{b}} \right)}^{d}}={{a}^{bd}}...................\left( 1 \right)$
Now, we will use the above formula to answer this question comfortably.
Now, let $a=4$ then, ${{a}^{2}}=16$ . Then, if we want to express ${{16}^{-2}}$ as a power with base 4 so, we will use the formula from the equation (1). Then,
\[\begin{align}
  & {{16}^{-2}}={{\left( {{4}^{2}} \right)}^{-2}} \\
 & \Rightarrow {{16}^{-2}}={{4}^{-2\times 2}} \\
 & \Rightarrow {{16}^{-2}}={{4}^{-4}} \\
\end{align}\]
Now, from the above result, we conclude that the value of ${{16}^{-2}}$ will be equal to ${{4}^{-4}}$ .
Thus, we can express ${{16}^{-2}}$ as a power with base in the form of ${{4}^{-4}}$ .
Hence, our final answer will be ${{4}^{-4}}$ .

Note: Here, the student should first try to understand what is asked in the question and then proceed in the right direction. After that, we should make use of the one of the basic concept of exponents that if ${{a}^{b}}=c$ then, ${{c}^{d}}={{\left( {{a}^{b}} \right)}^{d}}={{a}^{bd}}$ directly. Moreover, though the problem is very easy, we should always solve these types of questions by basic concepts of exponents to strengthen our concepts which will be further useful for solving tough problems.