Question

What is the unit digit in ${{7}^{105}}$
A.1
B.5
C.7
D.9

Answer 1

Hint: The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $2\times 2\times 2$ can be expressed as ${{2}^{3}}$. Here, the number two is called the base and the exponent represents the number of times the base is used as the factor.

Complete step-by-step answer:
By the property of the Power rule: to raise Power to power you have to multiply the exponents such as - ${{\left( {{2}^{a}} \right)}^{b}}={{2}^{ab}}$ and also, apply the product rule to multiply the exponents with the same base, you have to simply add the power such as ${{x}^{m}}\times {{x}^{n}}={{x}^{m+n}}$
Here we will split the power using the above two rules -
$\begin{align}
\Rightarrow & 105=4(26)+1 \\
 & {{7}^{105}}={{7}^{4(26)}}\times {{7}^{1}} \\
\end{align}$
$\Rightarrow$ Now, ${{7}^{4}}=2401$
Unit place is one.
 As a result the unit place of $\Rightarrow {{7}^{4(26)}}=1$
Now the required unit place of
$\Rightarrow {{7}^{105}}={{7}^{4(26)}}\times {{7}^{1}}=1\times 7=7$
Therefore, the required answer is – the unit digit in ${{7}^{105}}\,\text{is 7}$

So, the correct answer is “Option C”.

Note: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
I.Product of powers rule
II.Quotient of powers rule
III.Power of a power rule
IV.Power of a product rule
V.Power of a quotient rule
VI.Zero power rule
VII.Negative exponent rule