Question

How many digits are in \[{2^8} \times {5^{10}}\], when written out completely?
A.8
B.10
C.13
D.18
E.80

Answer 1

Hint: Here, we will use the exponential formulas to further write the given exponential numbers in a way such that we get a number 10 with any power. Then, we will count the number of digits to find the required answer.

Formula Used:
 We will use the following formulas:
1.\[{a^m} \cdot {a^n} = {a^{m + n}}\]
2.\[{a^m} \times {b^m} = {\left( {a \cdot b} \right)^m}\]

Complete step-by-step answer:
Given exponential numbers are: \[{2^8} \times {5^{10}}\]
We can also write the given number as:
\[{2^8} \times {5^{10}} = {2^8} \times {5^{8 + 2}}\]
Now, using the formula \[{a^m} \cdot {a^n} = {a^{m + n}}\] , we get
\[ \Rightarrow {2^8} \times {5^{10}} = {2^8} \times {5^8} \times {5^2}\]
Now, we are having the same power 8 with two different bases 2 and 5.
Hence, using the formula \[{a^m} \times {b^m} = {\left( {a \cdot b} \right)^m}\], we get
\[ \Rightarrow {2^8} \times {5^{10}} = {\left( {2 \times 5} \right)^8} \times {5^2}\]
Multiplying the terms inside the brackets, we get
\[ \Rightarrow {2^8} \times {5^{10}} = {10^8} \times {5^2}\]
Now, we can further write this as:
\[ \Rightarrow {2^8} \times {5^{10}} = 25 \times {10^8}\]
Since, the power of 10 is 8, this means that 10 is multiplied by itself 8 times.
Thus, we can write the product of given numbers as:
\[ \Rightarrow {2^8} \times {5^{10}} = 25 \times 100000000\]
Now multiplying the terms, we get
\[ \Rightarrow {2^8} \times {5^{10}} = 2500000000\]
Here, counting the number of digits, we get, one 2, one 5 and 8 zeros as \[\left( {1 + 1 + 8} \right) = 10\]digits
Thus, when written out completely, \[{2^8} \times {5^{10}}\] has 10 digits.
Therefore, option B is the correct answer.

Note: Exponential notation is a way of expressing numbers in the form of \[{a^n}\] which shows that the number in the base i.e. \[a\] has to be multiplied by itself \[n\] times. The exponential properties play an important role for solving the exponential numbers. For e.g., in this question when we saw numbers with different base but same power, we used the property: \[{a^m} \times {b^m} = {\left( {a \cdot b} \right)^m}\]. Hence, these properties help us to solve the question accurately and in less time.