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 $1$ $2, 4, 8, 6$
Predict the ones digit in the following:
$9^{20}$

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Solution

We, can see from table that after every $2^{n d}$ power unit place of $9$ is repeating
Or we can also say that even power of $9$ is $1$ and odd power is $9$
And, $20$ is even number.
So, $9^{20}$ has unit place $1$

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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(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.
$(i) 4^{12}$
$(i i) 9^{20}$
$(i i i) 3^{17}$
$(i v) 5^{100}$
$(v) 10^{500}$

Solve :

$(i) \frac{1}{3} x - 6 = \frac{5}{2} (i i) \frac{2 x}{3} - \frac{3 x}{8} = \frac{7}{12} (i i i) (x + 2) (x + 3) + (x - 3) (x - 2) - 2 x (x + 1) = 0 (i v) \frac{1}{10} - \frac{7}{x} = 35 (v) 13 (x - 4) - 3 (x - 9) - 4 (x + 4) = 0 (v i) x + 7 - \frac{8 x}{3} = \frac{17 x}{6} - \frac{5 x}{8} (v i i) \frac{3 x - 2}{4} - \frac{2 x + 3}{3} = \frac{2}{3} - x (v i i i) \frac{x + 2}{6} - (\frac{11 - x}{3} - \frac{1}{4}) = \frac{3 x - 4}{12} (i x) \frac{2}{5 x} - \frac{5}{3 x} = \frac{1}{15} (x) \frac{x + 2}{3} - \frac{x + 1}{5} = \frac{x - 3}{4} - 1 (x i) \frac{3 x - 2}{3} + \frac{2 x + 3}{2} = x + \frac{7}{6} (x i i) x - \frac{x - 1}{2} = 1 - \frac{x - 2}{3} (x i i i) \frac{9 x + 7}{2} - (x - \frac{x - 2}{7}) = 36 (x i v) \frac{6 x + 1}{2} + 1 = \frac{7 x - 3}{3}$

Solve the equations:

(i) 5x = 3x + 24;

(ii) 8t + 5 = 2t − 31;

(iii) 7x − 10 = 4x + 11;

(iv) 4z + 3 = 6 + 2z;

(v) 2x − 1 = 14 − x;

(vi) 6x + 1 = 3(x − 1) + 7;

(vii) ;

(viii) ;

(ix) 3(x + 1) = 12 + 4 (x − 1);

(x) 2x − 5 = 3(x − 5);

(xi) 6(1 − 4x) + 7(2 + 5x) = 53;

(xii) 3(x + 6) + 2 (x + 3) = 64;

(xiii) ;

(xiv) .

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

Predict the ones digit, copy and complete this table and answer the question that follow.X1x2x3x4x5x6x7x8x9x10x1122143184116513261647112881256Ones Digits of the Powers12,4,8,6Predict the ones digit in the following:920

We, can see from table that after every 2nd power unit place of 9 is repeating Or we can also say that even power of 9 is 1 and odd power is 9And, 20 is even number.So, 920 has unit place 1

We, can see from table that after every $2^{n d}$ power unit place of $9$ is repeating
Or we can also say that even power of $9$ is $1$ and odd power is $9$
And, $20$ is even number.
So, $9^{20}$ has unit place $1$