Express the following with rational denominator: 16 41 - 5

Rationalizing the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction (denominator) to the top of the fraction (numerator). Multiply both numerator and denominator by conjugate of denominator 41 + 5 , we get 16 41 - 5 × 41 + 5 41 + 5 As we know that a + b a - b = a 2 - b 2 ⇒ 16 41 + 5 41 2 - 5 2 ⇒ 16 41 + 5 41 - 25 ⇒ 16 41 + 5 16 ⇒ 41 + 5

Represent 5.5 on the real line.

(i) Draw a line segment A B = 5 . 5 and extend it to C such that B C = 1 . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 5 . 5 and point E represent 5 . 5 Proof: A B = 5 .5 c m , B C = 1 c m , A C = 5 .5 + 1 = 6 .5 c m O A = O B = 6 .5 2 = 3 . 25 c m O B = O C - B C = 3 . 25 - 1 = 2 . 25 c m In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 3 . 25 2 - 2 . 25 2 = 10 . 5625 - 5 . 0625 ⇒ B D 2 = 5 .5 ⇒ B D = 5 .5 ∴ B D = B E = 5 .5 c m and point E represents 5 . 5

If x = 3 + 8 , show that x 2 + 1 x 2 = 34

Given, x = 3 + 8 So, ⇒ 1 x = 1 3 + 8 ⇒ 1 x = 1 3 + 8 × 3 - 8 3 - 8 = 3 - 8 3 2 - 8 2 = 3 - 8 9 - 8 = 3 - 8 1 = 3 - 8 Now, x 2 = 3 + 8 2 = 9 + 8 + 2 × 3 × 8 = 17 + 6 8 1 x 2 = 3 - 8 2 = 9 + 8 - 2 × 3 × 8 = 17 - 6 8 Now, x 2 + 1 x 2 = 17 + 6 8 + 17 - 6 8 = 17 + 17 ⇒ x 2 + 1 x 2 = 34 Hence, proved.

Find the value of a and b if 3 + 7 3 - 7 = a + b 7

Given, 3 + 7 3 - 7 Rationalizing 3 + 7 3 - 7 = 3 + 7 3 + 7 3 - 7 3 + 7 = 3 + 7 2 3 2 - 7 2 As we know that a + b 2 = a 2 + 2 a b + b 2 = 3 2 + 7 2 + 2 × 3 × 7 2 = 9 + 7 + 6 7 2 = 16 + 6 7 2 = 2 8 + 3 7 2 = 8 + 3 7 Comparing RHS a + b 7 = 16 . . . ( 1 ) On equating rational and irrational part in ( 1 ) on both sides we get a = 8 , b = 3