\(\cfrac { { 9 }^{ 2 }\times { 7 }^{ 2 }\times { 2 }^{ 5 } }{ { 84 }^{ 3 } } \) = \(\cfrac { \left( { 3 }^{ 2 } \right) ^{ 2 }\times { 7 }^{ 3 }\times { 2 }^{ 5 } }{ \left( { 2 }^{ 2 }\times 3\times 7 \right) ^{ 3 } } \) 
= \(\cfrac { { 3 }^{ 4 }\times { 7 }^{ 3 }\times { 2 }^{ 5 } }{ { 2 }^{ 6 }\times { 3 }^{ 3 }\times { 7 }^{ 3 } } \) 
= \(\cfrac { { 3 }^{ 4-3 }\times { 7 }^{ 3-3 } }{ { 2 }^{ 6-5 } } \) 
= \(\cfrac { { 3 }^{ 1 }\times { 7 }^{ 0 } }{ { 2 }^{ 1 } } =\cfrac { 3\times 1 }{ 2 } =\cfrac { 3 }{ 2 } \)

\(\left( { 5 }^{ o }+{ 6 }^{ -1 } \right) \times { 3 }^{ 3 }=\left( 1+\cfrac { 1 }{ 6 } \right) \times { 3 }^{ 3 }\) 
  

= \(\cfrac { 63 }{ 2 } \)

\(\left( -\cfrac { 1 }{ 3 } \right) ^{ -2 }\) 
\(\left( -\cfrac { 1 }{ 3 } \right) ^{ -2 }=-\cfrac { 1^{ -2 } }{ { 3 }^{ -2 } } =\cfrac { { -3 }^{ 2 } }{ { 1 }^{ 2 } } =-\cfrac { 9 }{ 1 } =-9\)

200 = 2 x 2 x 2 x 5 x 5
We have one triplets. We need one more 5 to get another triplet.
5 should be multiplied to make 200 a perfect cube.

729 = 3⁶
\(\sqrt [ 3 ]{ 729 } =\left( { 3 }^{ 6 } \right) ^{ 1/3 }={ 3 }^{ 2 }=9\) 
6859 = 19³
\(\sqrt [ 3 ]{ 6859 } =\left( { 19 }^{ 3 } \right) ^{ 1/3 }=19\)

10985 = 13 x 13 x 13 x 5
We have one triplets.
5 is left over.
The smallest number by which 10985 should be divided is 5 so that the quotient is a perfect cube.

Pythagorean triplet (2m, m² -1, m²+ 1)
The largest member m²+ 1 = 65 = 64 + 1
= 8² + 1
m = 8
2m = 2 x 8 = 16
m² - 1 = 8²- 1 = 64 - 1 = 63
The Pythagorean triplet (16, 63, 65)
(ii) Smallest member is to
Pythagorean triplet (2m, m² -1, m²+ 1)
The smallest member 2m = 10
m = 0 = 5
m² - 1 = 5²- 1 = 25 - 1 = 24
m² + 1 = 52+ 1 = 25 + 1 = 26
The Pythagorean triplet (10, 24, 26)

(i)
\(\left( \cfrac { n+1 }{ 2 } \right) ^{ 2 }={ 10 }^{ 2 }\) 
\(\cfrac { n+1 }{ 2 } =10\) 
n + 1 = 20
n = 20 - 1 = 19
The sum of consecutive odd natural numbers is,
1 + 3 + 5 + 7 +.... + 19
(ii) 11²
\(\left( \cfrac { n+1 }{ 2 } \right) ^{ 2 }={ 11 }^{ 2 }\) 
\(\cfrac { n+1 }{ 2 } =11\) 
n + 1 = 22
n = 22 - 1 = 21
The sum of consecutive odd natural numbers is,
1 + 3 + 5 + 7 + .... + 21

\(\cfrac { { m }^{ 2 }-1 }{ 2 } ,\cfrac { { m }^{ 2 }+1 }{ 2 } \) 
\(\cfrac { { 15 }^{ 2 }-1 }{ 2 } ,\cfrac { { 15 }^{ 2 }+1 }{ 2 } \) 

\(\cfrac { 225-1 }{ 2 } ,\cfrac { 225+1 }{ 2 } \) 

15² = 112 + 113
(ii)
\(\cfrac { { m }^{ 2 }-1 }{ 2 } ,\cfrac { { m }^{ 2 }+1 }{ 2 } \) 
\(\cfrac { { 19 }^{ 2 }-1 }{ 2 } ,\cfrac { { 19 }^{ 2 }+1 }{ 2 } \) 
\(\cfrac { 361-1 }{ 2 } ,\cfrac { 361+1 }{ 2 } \) 

19² = 180 + 181

(i) 144
Step 1 : 144 - 1 = 143
Step 2 : 143 - 3 = 140
Step 3 : 140 - 5 =135
Step 4 : 135 - 7 = 128
Step 5 : 128 - 9 = 119
Step 6 : 119 - 11 = 108
Step 7 : 108 - 13 = 95
Step 8 : 95 - 15 = 80
Step 9 : 80 - 17 = 63
Step 10 : 63 - 19 = 44
Step 11 : 44 - 21 = 23
Step 12 : 23 - 23 = 0
\(\therefore \sqrt { 144 } =12\) 
(ii) Step 1 : 256 - 1 = 255
Step 2 : 255 -'-3 = 252
Step 3 : 252 - 5 = 247
Step 4 : 247 - 7 = 240
Step 5 : 240 - 9 = 231
Step 6 : 231-11 = 220
Step 7 : 220 - 13 = 207
Step 8 : 207 - 15 = 192
Step 9 : 192 - 17 = 175
Step 10 : 175 -19 = 156
Step 11 : 156 - 21 = 135
Step 12 : 135 - 23 = 112
Step 13 : 112 - 25 = 87
Step 14 : 87 - 27 = 60
Step 15 : 60 - 29 = 31
Step 16 : 31 - 31 = 0
\(\therefore \sqrt { 256 } =16\) 
(iii) 784
Step 1 : 784 - 1 = 783
Step 2 : 783 - 3 = 780
Step 3 : 780 - 5 = 775
Step 4 : 775 - 7 = 768
Step 5 : 768 - 9 = 759
Step 6 : 759 - 11 = 748
Step 7 : 748 - 13 = 735
Step 8 : 735 - 15 = 720
Step 9 : 720 - 17 = 703
Step 10 : 703 - 19 = 684
Step 11 : 684 - 21 = 663 .
Step 12 : 663 - 23 = 640
Step 13 : 640 - 25 = 615
Step 14 : 615 - 27 = 588
Step 15 : 588 - 29 = 559
Step 16 : 559 - 31 = 528
Step 17 : 528 - 33 = 495
Step 18 : 495 - 35 = 460
Step 19 : 460 - 37 = 423
Step 20 : 423 - 39 = 384
Step 21 : 384 - 41 = 343
Step 22 : 343 - 43 = 300
Step 23 : 300 - 45 = 255
Step 24 : 255 - 47 = 208
Step 25 : 208 - 49 = 159
Step 26 : 159 - 51 = 108
Step27 : 108 - 53 = 55
Step 28 : 55 - 55 = 0
\(\therefore \sqrt { 784 } =28\)