Question

Write the following numbers in powers of \[2\].
(i) 2
(ii) 8
(iii) 32
(iv) 128
(v) 256

Answer 1

Hint: Here, we will write the given numbers in the powers of 2. We will find the factors of the given numbers by using the prime factorization and by using the factors, we will write the given numbers in the powers of 2. Thus, the required answer will be in powers of 2.

Complete Step by Step Solution:
We are given the numbers 2, 8, 32, 128, 256.
We will write the numbers in the powers of \[2\].
\[2\]
We will write the given number in the form of factors by using the method of Prime factorization.
\[ \Rightarrow \] \[\begin{array}{l}2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, \[2\] can be written as \[2\] and in the powers of \[2\] as \[{2^1}\].
\[8\]
We will write the given number in the form of factors by using the method of Prime factorization.
\[\begin{array}{l}2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, \[8\] can be written as \[2 \times 2 \times 2\] and in the powers of \[2\]as\[{2^3}\].
\[32\]
We will write the given number in the form of factors by using the method of Prime factorization.
\[\begin{array}{l}2\left| \!{\underline {\,
  {32} \,}} \right. \\2\left| \!{\underline {\,
  {16} \,}} \right. \\2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, \[32\] can be written as \[2 \times 2 \times 2 \times 2 \times 2\] and in the powers of \[2\] as \[{2^5}\].
\[128\]
We will write the given number in the form of factors by using the method of Prime factorization.
\[\begin{array}{l}2\left| \!{\underline {\,
  {128} \,}} \right. \\2\left| \!{\underline {\,
  {64} \,}} \right. \\2\left| \!{\underline {\,
  {32} \,}} \right. \\2\left| \!{\underline {\,
  {16} \,}} \right. \\2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, \[128\] can be written as \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\] and in the powers of \[2\] as \[{2^7}\].
\[256\]
We will write the given number in the form of factors by using the method of Prime factorization.
\[\begin{array}{l}2\left| \!{\underline {\,
  {256} \,}} \right. \\2\left| \!{\underline {\,
  {128} \,}} \right. \\2\left| \!{\underline {\,
  {64} \,}} \right. \\2\left| \!{\underline {\,
  {32} \,}} \right. \\2\left| \!{\underline {\,
  {16} \,}} \right. \\2\left| \!{\underline {\,
  8 \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \\{\rm{ }}\underline 1 \end{array}\]
Thus, \[256\] can be written as \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\] and in the powers of \[2\] as \[{2^8}\].
Therefore, \[2\] can be written in the powers of \[2\] as \[{2^1}\].
Therefore, \[8\] can be written in the powers of \[2\] as \[{2^3}\].
Therefore, \[32\] can be written in the powers of \[2\] as \[{2^5}\].
Therefore, \[128\] can be written in the powers of \[2\] as \[{2^7}\].
Therefore, \[256\] can be written in the powers of \[2\] as \[{2^8}\].

Note: We should remember that the prime factorization method is done only by using the Prime factor 2 since we will find the numbers in the powers of 2. Power is defined as the number which is represented in the form of repeated multiplication of value or an integer. We should be clear that if \[{a^n}\] , then it is said to be the power, and \[n\] is said to be the exponent of \[a\].