Question

The value of \[{{4}^{-3}}\]
$\begin{align}
  & a)-64 \\
 & b)64 \\
 & c)-12 \\
 & d)\dfrac{1}{64} \\
\end{align}$

Answer 1

Hint: Now we know that ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ . Hence using this we will first write \[{{4}^{-3}}\] with positive power. Now we also know that for positive m we have ${{a}^{m}}=a\times a\times a\times m\text{ times}$ . Hence we will use this to evaluate the obtained expression and hence find the value of \[{{4}^{-3}}\] .

Complete step by step answer:
Now let us first understand the concept of indices. An index is a number which is raised to a power. Consider the number ${{2}^{3}}$ here 3 is called the index number. Now let us understand the meaning of ${{a}^{m}}$ . ${{a}^{m}}=a\times a\times a\times ......m\text{ times}$ . Hence ${{a}^{m}}$ is nothing but just multiplication of a m times. Now with this we can expand ${{a}^{m}}$ for all powers which are natural numbers.
Now note that ${{a}^{0}}=1$ and when there is no power mentioned then the value power is assumed as 1. For example $2={{2}^{1}}$ .
Now let us see what happens when power is a fraction. Consider ${{a}^{\dfrac{p}{q}}}$ then we have ${{a}^{\dfrac{p}{q}}}=\sqrt[q]{{{a}^{p}}}$ .
Now the power of a number can also be negative. For example a number is written as ${{a}^{-n}}$ .
Now in this case ${{a}^{-n}}$ can be written as $\dfrac{1}{{{a}^{n}}}$ . For example ${{2}^{-3}}=\dfrac{1}{{{2}^{3}}}$ .
Now consider the given number \[{{4}^{-3}}\] .
Now since we have -3 as power we will make this positive by taking this to the denominator.
Hence we get ${{4}^{-3}}=\dfrac{1}{{{4}^{3}}}$ .
Now we have ${{4}^{3}}$ in the denominator.
Now we know that for positive integer m, ${{a}^{m}}=a\times a\times a.....m\text{ times}$
Hence ${{4}^{3}}=4\times 4\times 4$ .
Now we know that $4\times 4=16$
Hence ${{4}^{3}}=16\times 4$ . Now $16\times 4=64$ .
Hence we get ${{4}^{3}}=64$
Hence we have ${{4}^{-3}}=\dfrac{1}{{{4}^{3}}}=\dfrac{1}{64}$ .

So, the correct answer is “Option d”.

Note: Now we have some properties related to indices which we can use while solving questions related to indices. For example ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ , $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ and ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ . Now when the power is 2 we call it the square of a number and when the power is 3 we call it a cube of number.