Question

What is $2$ raised to the power of $100$ ?

Answer 1

Hint:Here in this question we have to find the value of 2 power 100. So first we write the number in the form of exponent and then using of laws of indices and tables of multiplication write the number as \[{\left( {{2^{10}}} \right)^{10}}\]. Then on multiplying the number 2 10 times by itself and the obtained result by itself another 10 times we obtain the result.

Complete step by step answer:
The exponential number is defined as the number of times the number is multiplied by itself. It is represented as \[{a^n}\], where $a$ is the numeral and $n$ represents the number of times the number is multiplied. Now we consider the given question. The number $2$ raised to the power of $100$. Usually, the power term is written like a superscript of the given number. For example, if we have a number \[a\] is raised to the power of $b$. This can be written as \[{a^b}\], here $b$ is the power so we have written it as a superscript of $a$.Therefore, the number $2$ raised to the power of $100$ is written as \[{2^{100}}\]. We solve this question by two methods.

Method 1: In this method we are going to solve this problem by considering the logarithmic function. Consider,
\[ \Rightarrow y = {2^{100}}\]
On applying the log on both sides we get
\[ \Rightarrow \log y = \log ({2^{100}})\]
By the laws of logarithmic functions \[\log ({a^m}) = m\log (a)\], the above inequality is written as
\[ \Rightarrow \log y = 100\log (2)\]
The \[\log 2 = 0.69315\], on substituting the value in above inequality we have
\[ \Rightarrow \log y = 100 \times 0.69315\]
On multiplying 100 and 0.69315 we get
\[ \Rightarrow \log y = 69.315\]
Taking antilog on both sides we get
\[ \Rightarrow {e^{\log y}} = {e^{69.315}}\]
On simplifying we get
\[ \therefore y = 1.2680081 \times {10^{30}}\]

Method 2: Consider
\[ \Rightarrow {2^{100}}\]
By the tables of multiplication, the number 100 can be written as a product of 10 and 10.
 \[ \Rightarrow {2^{10 \times 10}}\]
By the laws of indices, we have \[{a^{mn}} = {({a^m})^n}\], the above term can be written as
\[ \Rightarrow {\left( {{2^{10}}} \right)^{10}}\]
First, we simplify the term which is present in the bracket. When 2 is multiplied 10 times by itself we get the product as 1024. Therefore, we have
\[ \Rightarrow {\left( {1024} \right)^{10}}\]
The number 1024 has the power 10. So the number 1024 is multiplied 10 times by itself. When the number is multiplied by itself we get the product as
\[ \Rightarrow 1,267,650,600,228,229,401,496,703,205,376\]
This is a huge number, this can be written in the scientific form. So we have
\[ \therefore 1.2680081 \times {10^{30}}\]

Hence the 2 raised to the power of the 100 is \[1.2680081 \times {10^{30}}\].

Note:The exponential number is defined as the number of times the number is multiplied by itself. It is represented as \[{a^n}\], where a is the numeral and n represents the number of times the number is multiplied. For the exponential numbers we have a law of indices and by applying it we can solve the given number. A huge number can be written in the scientific form.