Question 128i Predicting the ones digit- copy and complete this table and answer the questions that follow- Powers Table Wegin array 1-1- - Wline2 - 1 - 4 - llthline - 1 - 8 - 11 hline4 - 1 - 16 - Nhline5 - 1 - 32 - 11 hline- 1 - 64 - Illhline 7 - 1 - 128 - 11 hline- 18256 - - lllhlinelendarray Describe patterns you see in the ones digits of the powers-

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Standard IX

Mathematics

Fundamental Laws of Logarithms

Question 128i...

Question

Question 128(i)

Predicting the ones digit, copy and complete this table and answer the questions that follow.
Powers Table
$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 2 & 1 & 4 3 & 1 & 8 4 & 1 & 16 5 & 1 & 32 6 & 1 & 64 7 & 1 & 128 8 & 1 & 256 Ones digits of the powers & 1 & 2, 4, 6, 8 \end{matrix}$
Describe patterns you see in the ones digits of the powers.

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Solution

On the basis of given pattern in $1^{x}$ and $2^{x}$ , we can make 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.

$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 2 & 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 3 & 1 & 8 & 27 & 64 & 125 & 216 & 343 & 512 & 729 & 1000 4 & 1 & 16 & 81 & 256 & 625 & 1296 & 2401 & 4096 & 6561 & 10000 5 & 1 & 32 & 243 & 1024 & 3125 & 7776 & 16807 & 32768 & 59049 & 100000 6 & 1 & 64 & 729 & 4096 & 15625 & 46656 & 117649 & 262144 & 531441 & 1000000 7 & 1 & 128 & 2187 & 16384 & 78125 & 279936 & 823543 & 2097152 & 4782969 & 10000000 8 & 1 & 256 & 6561 & 65536 & 390625 & 1679616 & 5764801 & 16777216 & 43046721 & 100000000 Ones digits of the powers & 1 & 2, 4, 6, 8 & 3, 9, 7, 1 & 4, 6 & 5 & 6 & 7, 9, 3, 1 & 8, 4, 2, 6 & 9, 1 & 0 \end{matrix}$

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Question 128(iii)

Predicting the ones digit, copy and complete this table and answer the questions that follow.

Powers Table
$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 2 & 1 & 4 3 & 1 & 8 4 & 1 & 16 5 & 1 & 32 6 & 1 & 64 7 & 1 & 128 8 & 1 & 256 O n e s d i g i t s o f t h e p o w e r s & 1 & 2, 4, 6, 8 \end{matrix}$

Predict the ones digit in the following.
$(i) 31^{10}$
$(i i) 12^{10}$
$(i i i) 17^{21}$
$(i v) 29^{10}$

Question 128(i)

Predicting the ones digit, copy and complete this table and answer the questions that follow.
Powers Table
$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 2 & 1 & 4 3 & 1 & 8 4 & 1 & 16 5 & 1 & 32 6 & 1 & 64 7 & 1 & 128 8 & 1 & 256 Ones digits of the powers & 1 & 2, 4, 6, 8 \end{matrix}$
Describe patterns you see in the ones digits of the powers.

Question 128(iii)

Predicting the ones digit, copy and complete this table and answer the questions that follow.

Powers Table
$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 2 & 1 & 4 3 & 1 & 8 4 & 1 & 16 5 & 1 & 32 6 & 1 & 64 7 & 1 & 128 8 & 1 & 256 O n e s d i g i t s o f t h e p o w e r s & 1 & 2, 4, 6, 8 \end{matrix}$

Predict the ones digit in the following.
$(i) 31^{10}$
$(i i) 12^{10}$
$(i i i) 17^{21}$
$(i v) 29^{10}$

Question 128(ii)

$\begin{matrix} X & 1^{x} & 2^{x} & 3^{x} & 4^{x} & 5^{x} & 6^{x} & 7^{x} & 8^{x} & 9^{x} & 10^{x} 1 & 1 & 2 2 & 1 & 4 3 & 1 & 8 4 & 1 & 16 5 & 1 & 32 6 & 1 & 64 7 & 1 & 128 8 & 1 & 256 Ones digits of the powers & 1 & 2, 4, 6, 8 \end{matrix}$
Predict the ones gigit in the following.
$(i) 4^{12}$
$(i i) 9^{20}$
$(i i i) 3^{17}$
$(i v) 5^{100}$
$(v) 10^{500}$

Complete the last column of the table:

$\begin{matrix} S.No. & Equation & Value & Say, whether the equation is satisfied (yes/no) (i) & x + 3 = 0 & x = 3 (i i) & x + 3 = 0 & x = 0 (i i i) & x + 3 = 0 & x = - 3 (i v) & x - 7 = 1 & x = 7 (v) & x - 7 = 1 & x = 8 (v i) & 5 x = 25 & x = 0 (v i i) & 5 x = 25 & x = 5 (v i i i) & 5 x = 25 & x = - 5 \end{matrix}$

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Question 128(i) Predicting the ones digit, copy and complete this table and answer the questions that follow. Powers Table X1x2x3x4x5x6x7x8x9x10x1122143184116513261647112881256Ones digits of the powers12,4,6,8 Describe patterns you see in the ones digits of the powers.