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Exponents and Powers — MATHS STD 8 — Question

Gujarat BoardEnglish MediumSTD 8MATHSExponents and Powers5 Marks
Question
Predicting the ones digit, copy and complete this table and answer the questions 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 Digit of the Powers
$1$
$2, 4, 8, 6$
 
 
 
 
 
 
 
 
$a.$ Describe patterns you see in the ones digits of the powers.
$b.$ Predict the ones digit in the following:
$i. 4^{12}$
$ii. 9^{20}$
$iii. 3^{17}$
$iv. 5^{100}$
$v. 10^{500}$
$c.$ Predict the ones digit in the following:
$i. 31^{10}$
$ii. 12^{10}$
$iii.17^{21}$
$iv. 29^{10}$
✓

Answer

$a.$ On the basis of given pattern in $1^x$ and $2^x$ , we can make more patterns 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 details about the patterns.
$x$
$1^x$
$2^x$
$3^x$
$4^x$
$5^x$
$6^x$
$7^x$
$8^x$
$9^x$
$10^x$
$1$
$1$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$2$
$1$
$4$
$9$
$16$
$25$
$36$
$49$
$64$
$81$
$100$
$3$
$1$
$8$
$27$
$64$
$125$
$216$
$343$
$512$
$729$
$1000$
$4$
$1$
$16$
$81$
$256$
$625$
$1296$
$2401$
$4096$
$6561$
$10000$
$5$
$1$
$32$
$243$
$1024$
$3125$
$7776$
$16807$
$32768$
$59049$
$100000$
$6$
$1$
$64$
$729$
$4096$
$15625$
$46656$
$117649$
$262144$
$531441$
$1000000$
$7$
$1$
$128$
$2187$
$16384$
$78125$
$279936$
$823543$
$2097152$
$4782969$
$10000000$
$8$
$1$
$256$
$6561$
$65536$
$390625$
$1679616$
$5764801$
$16777216$
$43046721$
$100000000$
Ones Digit of the Powers
$1$
$2, 4, 8, 6$
$3, 9, 7, 1$
$4, 6$
$5$
$6$
$7, 9, 3, 1$
$8, 4, 2, 6$
$9, 1$
 $0$
$b.$ Predict the ones digit:
$i.$ Ones digit in $4^{12}$ is $6$.
$ii.$ Ones digit in $9^{20}$ is $1$.
$iii.$ Ones digit in $3^{17}$ is $3$.
$iv.$ Ones digit in $5^{100}$ is $5$.
$v.$ Ones digit in $10^{500}$ is $0$.
$c.$ Predict the ones digit:
$i.$ Ones digit in $31^{10}$ is $1$.
$ii.$ Ones digit in $12^{10}$ is $4$.
$iii.$ Ones digit in $17^{21}$ is $7$.
$iv.$ Ones digit in $29^{10}$ is $1$.

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