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.