Question

The exponential form 768 is ${{2}^{8}}\times 3$.
(a) True
(b) False

Answer 1

Hint: Use the prime factorization method to write the given number 768 as the product of its prime factors. If the same factors are repeating then write them in exponential form. Now, compare the exponents of the factors 2 and 3. If they are 8 and 1 respectively then the statement is true otherwise false. If any factor other than 2 and 3 is present then also the statement becomes false.

Complete step-by-step solution:
Here we have been provided with the statement ‘the exponential form of 768 is ${{2}^{8}}\times 3$. We have to check if it is true or false.
Now, to check the correctness of the given statement we need to write the number 768 as the product of its prime factors. We know that any composite number can be written as the product of its prime factors. So using the prime factorization method we have,
\[\begin{align}
  & 2\left| \!{\underline {\,
  768 \,}} \right. \\
 & 2\left| \!{\underline {\,
  384 \,}} \right. \\
 & 2\left| \!{\underline {\,
  192 \,}} \right. \\
 & 2\left| \!{\underline {\,
  96 \,}} \right. \\
 & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & \,\,\,\,3 \\
\end{align}\]
Clearly we can see that 2 is appearing 8 time and 3 is appearing 1 time in the above prime factorization, so we can write,
$\Rightarrow 768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3$
In exponential form we can write,
\[\therefore 768={{2}^{8}}\times 3\]
Therefore, we can conclude that the given statement is correct that means it is true.
Hence, option (a) is the correct answer.

Note: Note that the prime factorization is very useful in calculating square roots of a number and to check if they are perfect squares or not. If all the factors can be written as the exponent of 2 then the number is a perfect square otherwise not. This method is also helpful while calculating the L.C.M and H.C.F of two or more numbers.