Question

What is the unit digit in ${(4137)^{754}}$?
A) $1$
B) $3$
C) $7$
D) $9$

Answer 1

Hint: The power is used to express mathematical value in the short form; it is an expression that signifies the repeated multiplication of the same factor. For example - $2 \times 2 \times 2$ can be stated as ${2^3}$. Here, the number two is named the base and the exponent signifies the number of times the base is used as the factor. Here the unit of the power and exponent is asked so here will consider the unit term of the base of the given number.

Complete step by step solution:
Given number: ${(4137)^{754}}$
The unit place of the above number of the base is: $7$
We can observe the base and the power for the term $7$
Unit digit when the term is ${7^1} = 7$
Unit digit when the term is ${7^2} = 9$
Unit digit when the term is ${7^3} = 3$
Unit digit when the term is ${7^4} = 1$
Unit digit when the term is ${7^5} = 7$
Hence the pattern for unit digit of $7$is always $7,9,3,1$
Now the given number is ${(4137)^{754}}$
Just to concentrate on the unit digit ${(7)^{754}} = {(7)^{2(377)}}$
Unit digit of ${7^2} = 9$
Hence, from the given multiple choices option (D) is the correct answer.

Note:
Always remember the basic and important seven rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules and remember it thoroughly which describes how to solve different types of exponents problems and how to use operators such as addition, subtraction, multiplication and division of the exponents.
> Product of powers rule
> Quotient of powers rule
> Power of a power rule
> Power of a product rule
> Power of a quotient rule
> Zero power rule
> Negative exponent rule