Question

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

Answer 1

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.