Carpenter A fits the parts of a chair in 1 minute = \(\cfrac { 1 }{ 15 } \)
Carpenter B fits the parts of a chaaiir In 1 minute = \(\cfrac { 1 }{ 18 } \) 
(A+B) fit the parts in 1 minute = \(\cfrac { 1 }{ 15 } +\cfrac { 1 }{ 18 } =\cfrac { 6+5 }{ 90 } =\cfrac { 11 }{ 90 } \) 
Time taken to fit. 1 chair = \(\cfrac { 90 }{ 11 } \) mtrs 

 Time taken to fit the parts for 22 chairs = 3 hours

(A + C)'s 1 day work = \(\cfrac { 1 }{ 6 } \) 
A's 1 day work = \(\cfrac { 1 }{ 12 } \) 
C's 1 day work = \(\cfrac { 1 }{ 6 } -\cfrac { 1 }{ 12 } =\cfrac { 2-1 }{ 12 } =\cfrac { 1 }{ 12 } \) 
(B + C)'s 1 day work = \(\cfrac { 1 }{ 3 } \) 
B's Iday work = \(\cfrac { 1 }{ 3 } -\cfrac { 1 }{ 12 } =\cfrac { 4-1 }{ 12 } =\cfrac { 3 }{ 12 } =\cfrac { 1 }{ 4 } \) 
B alone can complete the work in 4 hours.

Number of Soaps increase, so the days also increase.
So it is direct variation
The multiplying factor is \(\cfrac { 14400 }{ 9600 } \) 
Time increase days decrease
So it is indirect variation
The multiplying factor is \(\cfrac { 15 }{ 18 } \) 

It will produce 14400 soaps in 7^1/2th days.

Days increase. So men will decrease
So, it is inverse proportion
The multiplying factor is \(\cfrac { 18 }{ 20 } \) 
Time increase, so men will decrease.
So, it is inverse proportion
The multiplying factor is \(\cfrac { 12 }{ 14 } \) 
 
162 men are required to finish the job 20 days working 14 hours a day.

(3.769 x 10⁵) + 4.21 x 10⁵
3.769 x 10⁵ + 4.21 x 10⁵
(3.769 + 4.21)10⁵
7.979 x 10⁵

\(\sqrt { 286225 } \) 

\(\sqrt { 286225 } =535\) 
\(\sqrt { 2862.5 } =53.5\) 
\(\sqrt { 28.6225 } =5.35\) 
\(\sqrt { 2862.5 } +\sqrt { 28.6225 } =53.5+5.35\) 
= 58.85

Total number of cadets = 1536

If we take 39 x 39 cadets
15 cadets would be remained,
So square design should be 40 x 40
40 x 40 = 1600
. Number of cadets would be
required = 1600 - 1536 = 64

Area of the greeting card = 90cm²
= 9 x 10cm²
The length of its side are 9cm and 10 cm.

(625)^x = 15625
(25²)^x = (25)³
25^2x = 25³
2x = 3
x = 3/2
\({ x }^{ 2 }=\left( \cfrac { 3 }{ 2 } \right) ^{ 2 }=\cfrac { 9 }{ 4 } \) 
\({ x }^{ 3 }=\left( \cfrac { 3 }{ 2 } \right) ^{ 3 }=\cfrac { 27 }{ 8 } \) 
\({ x }^{ 2 }=\cfrac { 9 }{ 4 } \) 
\({ x }^{ 3 }=\cfrac { 27 }{ 8 } \) 

Total number of plants = 2401

Plants in one row = \(\sqrt { 2401 } \) 
=49
Number of plants in each row = 49
Number of rows = 49

Area of the square carpet = 1024 m²

The length of a si.de of the carpet = \(\sqrt { 1024 } \) 
= 32m
The length of a side of the carpet = 32m

\(\cfrac { { 10 }^{ x } }{ { 10 }^{ -3 } } ={ 1 }0^{ 9 }\) 

= 10^x+3 = 109
The bases are equal
Equate the exponents,
x+3 = 9
x = 9-3 = 6
x = 6

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

 \(\left( { 3 }^{ -1 }+{ 4 }^{ -2 }+{ 5 }^{ -3 } \right) ^{ 0 }=1\)

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

= \(\cfrac { 63 }{ 2 } \)

\(\left( \cfrac { { 2 }^{ -5 } }{ { 2 }^{ 7 } } \right) ={ 2 }^{ -2 }={ 2 }^{ -5 }\times { 2 }^{ -7 }\times { 2 }^{ -2 }\) 
\({ 2 }^{ -5-7-2 }={ 2 }^{ -14 }\) 
 

   \(\left( \cfrac { 1 }{ 2 } \right) ^{ -5 }=\cfrac { 1 }{ { 2 }^{ 3 } } ={ 2 }^{ 5 }=2\times 2\times 2\times 2\times 2=32\)

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.

\(\sqrt [ 3 ]{ 46656 } =\sqrt [ 3 ]{ { 2 }^{ 6 }\times { 3 }^{ 6 } } =\left( { 2 }^{ 6 }\times { 3 }^{ 6 } \right) ^{ 1/3 }\) 
= 2² \(\times\) 3²
Square root of \({ 2 }^{ 2 }\times { 3 }^{ 2 }\) is \(\sqrt { { 2 }^{ 2 }\times { 3 }^{ 2 } } \) 
= \(\left( { 2 }^{ 2 }\times { 3 }^{ 2 } \right) ^{ 1/6 }\) 
= 2 \(\times\) 3 = 6
The square root of cube root of 46656 is 6.

The two smallest perfect square numbers are 4 and 16
4 and 16 are perfect square.
And also 4 x 16 = 64 is a perfect cube.

(i)  \(\cfrac { 144 }{ 225 } \) 
\(\sqrt { \cfrac { 144 }{ 225 } } =\sqrt { \cfrac { { 12 }^{ 2 } }{ { 15 }^{ 2 } } } =\cfrac { 12 }{ 15 } \) 
\(\sqrt { \cfrac { 144 }{ 225 } } =\cfrac { 12 }{ 15 } \) 
(ii)
\(7\cfrac { 18 }{ 49 } \) 
 \(\sqrt { 7\cfrac { 18 }{ 49 } } =\sqrt { \cfrac { 361 }{ 49 } } =\sqrt { \cfrac { { 19 }^{ 2 } }{ { 7 }^{ 2 } } } =\cfrac { 19 }{ 7 } =2\cfrac { 5 }{ 7 } \) 
\(\sqrt { 7\cfrac { 18 }{ 49 } } =2\cfrac { 5 }{ 7 } \) 
(iii)
\(6\cfrac { 1 }{ 4 } \) 
\(\sqrt { 6\cfrac { 1 }{ 4 } } =\sqrt { \cfrac { 25 }{ 4 } } =\sqrt { \cfrac { { 5 }^{ 2 } }{ { 2 }^{ 2 } } } =\cfrac { 5 }{ 2 } =2\cfrac { 1 }{ 2 } \) 
\(\sqrt { 6\cfrac { 1 }{ 4 } } =2\cfrac { 1 }{ 2 } \) 

(iv) \(4\cfrac { 25 }{ 36 } \)
 \(\sqrt { 4\cfrac { 25 }{ 36 } } =\sqrt { \cfrac { 169 }{ 36 } } =\sqrt { \cfrac { { 13 }^{ 2 } }{ { 6 }^{ 2 } } } =\cfrac { 13 }{ 6 } =2\cfrac { 1 }{ 6 } \) 
= \(\sqrt { 4\cfrac { 25 }{ 36 } } =2\cfrac { 1 }{ 6 } \)
 

The given numbers are 8, 12 and 15
L.C.M of 8, 12 and 15

L.C.M = 2 x 2 x 2 x 3 x 5
= 2² X 2 x 3 x 5 = 120
The prime factors 2,3 and 5 do not have a pairs.
So we can multiply 120 by 2 x 3 x 5
120 x 2 x 3 x 5 = 3600 = 2² x i X 3² X 5²
So 3600 is the square number divisible by 8, 12 and 15.

(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

(i)

725 = 5 x 5 x 29
= 52 x 29
The second prime number 29 does not have a pair. Hence 725 is not a perfect square number.
(ii)

190 = 2 x 5 x 19
All the three prime numbers do not have pairs.
Hence 190 is not a perfect square number.
(iii) 841

841 = 29 x 29 = 29²
\(\sqrt { 841 } =\sqrt { 29^{ 2 } } =29\) 
\(\sqrt { 841 } =29\) 
Hence 841 is a perfect square number.
(iv) 1089
 
1089 = 3 x 3 x 11 x 11
= 3²X11²
\(\sqrt { 1089 } =\sqrt { { 3 }^{ 2 }\times { 11 }^{ 2 } } \) 
= \(\sqrt { (3\times 11)^{ 2 } } \) 
= 3 x 11
= 33
\(\sqrt { 1089 } =33\) 
Hence 1089 is a perfect square number.

(i)


\(\sqrt { 17956 } =134\) 
(ii) \(\sqrt { 11025 } \) 

(ii)

(iv)
(i
(v)