Question

Express 512 in exponential form.

Answer 1

Hint:
Here, we will first find the prime factors of the given number using factorization. Then, we will represent the prime factors in terms of product and count the number of times that particular number is being multiplied. Hence, we will be able to find the required exponential form by putting the number as the base and the number of times it is being repeated as the power.

Complete step by step solution:
In order to convert the given number into exponential form, first of all, we are required to find the prime factors of the given number.
Now, factorization is a method of writing an original number as the product of its various factors.
Also, prime numbers are those numbers which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.
Therefore, prime factorization of 512 is:
$\begin{array}{*{20}{l}}
  2&| & {512} \\
\hline
  2&| & {256} \\
\hline
  2&| & {128} \\
\hline
  2&| & {64} \\
\hline
  2&| & {32} \\
\hline
  2&| & {16} \\
\hline
  2&| & 8 \\
\hline
  2&| & 4 \\
\hline
  2&| & 2 \\
\hline
  {}&| & 1
\end{array}$
Hence, 512 can be written as:
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
Now, in order to express this in the exponential form, as we can see the number 2 is multiplying with itself 9 times.
Hence, the base of the exponential form will be 2, whereas, since it is multiplying 9 times, hence, we will take the power as 9.
Thus, the given number can be written in the exponential form as:
$512 = {2^9}$
Thus, this is the required answer.

Note:
An expression which represents the repeated multiplication of a same number is known as power. Whereas, when a number is written with a power then the power becomes the exponent of that particular number. It shows how many times that particular number will be multiplied by itself. Hence, whenever we are given the multiplication of the same numbers then, we can express that number with an exponent.