Question

Predict the ones digit, copy and complete this table and answer the question that follow.

$X$	$1^x$	$2^x$	$3^x$	$4^x$	$5^x$	$6^x$	$7^x$	$8^x$	$9^x$	${10}^x$
$1$	$1$	$2$
$2$	$1$	$4$
$3$	$1$	$8$
$4$	$1$	$16$
$5$	$1$	$32$
$6$	$1$	$64$
$7$	$1$	$128$
$8$	$1$	$256$
Ones Digits of the Powers	$1$	$2,4,8,6$

Describe patterns you see in the ones digit of the powers.

Answer 1

unit place of a product is obtained by multiplying unit places of both multipliers$

That is, $\text{(unit place of first number) x (unit place of second number) =unit place of product}$

We, will use this method to find unit places of number.

Ones digit that is unit place of $a^x$ repeats itself after some values of $x$. Then onward it is just repeat of ones place set till then.

So, we will find pattern of one's place and then write all ones place for every number.

For finding pattern we will multiply unit place in previous step with given number again, to find unit place of this step.

On the basis of given pattern in $1^x$ and $2^x$, we can make more patters for $3^x,4^x,5^x,6^x,7^x,8^x,9^x,{10}^x$.

Thus we, have following table which shows all unit place details about the given pattern-

x	$1^x$	$2^x$	$3^x$	$4^x$	$5^x$	$6^x$	$7^x$	$8^x$	$9^x$	${10}^x$
1	1	2	3	4	5	6	7	8	9	0
2	1	4	9	6	5	6	9	4	1	0
3	1	8	7	4	5	6	3	2	9	0
4	1	6	1	6	5	6	1	6	1	0
5	1	2	3	4	5	6	7	8	9	0
6	1	4	9	6	5	6	9	4	1	0
7	1	8	7	4	5	6	3	2	9	0
8	1	6	1	6	5	6	1	6	1	0