Find the multiplicative inverse of 3 5 - 3 ÷ 5 3 5

Given: 3 5 - 3 ÷ 5 3 5 By using the laws of exponents: a - n = 1 a n Thus, = 1 3 5 3 ÷ 5 3 5 = 5 3 3 ÷ 5 3 5 By using law: a m ÷ a n = a m - n = 5 3 3 - 5 = 5 3 - 2 The multiplicative inverse of a n is a - n . a n . a - n = 1 Thus, the multiplicative inverse of 5 3 - 2 is 5 3 2 Hence, the required answer is 5 3 2

Solve for x : 7 3 7 x - 2 = 7 7

Given: 7 3 7 x - 2 = 7 7 By using the laws of exponents: a m a n = a m - n ⇒ 7 3 - x - 2 = 7 7 ⇒ 7 3 - x - - 2 = 7 7 ⇒ 7 3 - x + 2 = 7 7 ⇒ 7 5 - x = 7 7 When bases are equal, then powers are also equal. ⇒ 5 - x = 7 ⇒ - x = 7 - 5 ⇒ - x = 2 ∴ x = - 2 Hence, the value of x is - 2 .

Simplify: 15 x 5 y - 2 z 4 5 x 3 y - 3 z 2

Given: 15 x 5 y - 2 z 4 5 x 3 y - 3 z 2 It can be written as follows: 15 5 × x 5 x 3 × y - 2 y - 3 × z 4 z 2 By using law: a m ÷ a n = a m - n = 3 × x 5 - 3 × y - 2 - - 3 × z 4 - 2 = 3 × x 2 × y 1 × z 2 = 3 x 2 y z 2 Hence, the required answer is 3 x 2 y z 2 .

Simplify: 42 x 3 y 2 z 4 6 x y 2 z 3

Given: 42 x 3 y 2 z 4 6 x y 2 z 3 The above value can be written as follows: 42 6 × x 3 x × y 2 y 2 × z 4 z 3 7 × x 3 x × z 4 z 3 By using the laws of exponents: a m a n = a m - n = 7 × x 3 - 1 × z 4 - 3 = 7 × x 2 × z 1 = 7 x 2 z Hence, the required answer is 7 x 2 z .

Simplify: 3 x - 2 y - 3 z 9 x - 2 y - 4

Given: 3 x - 2 y - 3 z 9 x - 2 y - 4 We can write the above value as follows: 3 9 × x - 2 x - 2 × y - 3 y - 4 × z By using the laws of exponents: a m a n = a m - n = 1 3 × y - 3 - - 4 × z = 1 3 × y - 3 + 4 × z = 1 3 y z = y z 3 Hence, the required answer is y z 3 .

Simplify: 42 x 3 y 2 z 4 16 x 2 y 3 z 8

Given: 42 x 3 y 2 z 4 16 x 2 y 3 z 8 We can write the given value as follows: = 42 16 × x 3 x 2 × y 2 y 3 × z 4 z 8 By using the laws of exponents: a m a n = a m - n = 21 8 x 3 - 2 . y 2 - 3 . z 4 - 8 = 21 8 x 1 y - 1 z - 4 By using the rule: a - n = 1 a n = 21 8 x × 1 y 1 × 1 z 4 = 21 8 x y z 4 = 21 x 8 y z 4 Hence, the required answer is 21 x 8 y z 4 .