Verify the following: - 4 5 × - 2 3 = - 2 3 × - 4 5

Given, - 4 5 × - 2 3 = - 2 3 × - 4 5 We know that, Product of rational numbers = Product of numerators Product of denominators Therefore, LHS = - 4 5 × - 2 3 = - 4 × - 2 5 × 3 = 8 15 RHS = - 2 3 × - 4 5 = - 2 × - 4 3 × 5 = 8 15 ∴ LHS = RHS Therefore, - 4 5 × - 2 3 = - 2 3 × - 4 5 The above verifies the commutative property of rational numbers. Hence, verified.

If a = 1 5 , b = 1 3 and c = - 2 5 , then verify that a × b × c = a × ( b × c ) .

Given, a = 1 5 , b = 1 3 and c = - 2 5 . Therefore, LHS = a × b × c = 1 5 × 1 3 × - 2 5 = 1 15 × - 2 5 = - 2 75 RHS = a × ( b × c ) = 1 5 × 1 3 × - 2 5 = 1 5 × - 2 15 = - 2 75 ∴ a × b × c = a × b × c = - 2 75 The above verifies the associative property of rational numbers. Hence, verified.

Is 8 9 the multiplicative inverse of - 1 1 8 ? Why or why not?

Given numbers are 8 9 and - 1 1 8 . - 1 1 8 is equal to - 9 8 . And, - 9 8 × 8 9 = - 1 A multiplicative inverse or reciprocal for a number x , denoted by 1 x , is a number which when multiplied by x yields the multiplicative identity 1 . But, here we observe that the product is not 1 . Hence, 8 9 is not the multiplicative inverse of - 1 1 8 .

Verify the following: 2 3 + 4 5 is a rational number

Given, 2 3 + 4 5 = 2 3 + 4 5 = 10 + 12 15 = 22 15 We know that, A rational number is a number expressed in the form p q where the denominator, q ≠ 0 . In 22 15 , the denominator is not equal 0 . Therefore, 2 3 + 4 5 is a rational number.

Verify the following: 3 7 × 5 9 × 2 5 = 3 7 × ( 5 9 × 2 5 )

Given: 3 7 × 5 9 × 2 5 = 3 7 × 5 9 × 2 5 LHS ⇒ 3 7 × 5 9 3 × 2 5 = 5 21 × 2 5 = 2 21 RHS ⇒ 3 7 × 5 9 × 2 5 = 3 7 × 2 9 3 = 2 21 ⇒ LHS = RHS Therefore, 3 7 × 5 9 × 2 5 = 3 7 × 5 9 × 2 5 Hence, verified.

Verify the following: 7 2 × 3 5 = 3 5 × 7 2

Given, 7 2 × 3 5 = 3 5 × 7 2 We know that, Product of rational numbers = Product of numerators Product of denominators Therefore, LHS = 7 2 × 3 5 = 7 × 3 2 × 5 = 21 10 RHS = 3 5 × 7 2 = 3 × 7 5 × 2 = 21 10 ⇒ LHS = RHS Therefore, 7 2 × 3 5 = 3 5 × 7 2 The above verifies the commutative property of rational numbers. Hence, verified.

Is 0 . 3 the multiplicative inverse of 3 1 3 ? Why or why not ?

Given numbers are 0 . 3 and 3 1 3 . 0.3 = 3 10 3 1 3 = 10 3 A multiplicative inverse or reciprocal for a number x , denoted by 1 x , is a number which when multiplied by x yields the multiplicative identity 1 As we know, multiplicative inverse of a b is b a , multiplicative inverse of 10 3 is 3 10 . Hence, 0 . 3 is the multiplicative inverse of 3 1 3 .

Verify the following: - 3 7 + 5 2 + 5 6 = - 3 7 + 5 2 + 5 6

Given, - 3 7 + 5 2 + 5 6 = - 3 7 + 5 2 + 5 6 According to associative property of rational numbers in addition, for any three rational numbers a , b and c , a + b + c = a + b + c . Therefore, for a = - 3 7 , b = 5 2 and c = 5 6 , then - 3 7 + 5 2 + 5 6 = - 3 7 + 5 2 + 5 6 follows the associative property of rational numbers in addition as it is in the form a + b + c = a + b + c . Therefore, - 3 7 + 5 2 + 5 6 = - 3 7 + 5 2 + 5 6 Hence, verified.

Verify that – – x = x for x = 11 15

Given, x = 11 15 . The additive inverse of x is - x ..... (as x + - x = 0 ) The same for 11 15 + - 11 15 = 0 , shows that the additive inverse of - 11 15 is 11 15 Or, - - 11 15 = 11 15 i.e. – – x = x Hence, it is verified that – – x = x

Verify: a + b = b + a when a = - 3 7 and b = 2 3 .

Given, a = - 3 7 and b = 2 3 Putting the values of a and b , we get LHS ⇒ a + b = - 3 7 + 2 3 = - 9 + 14 21 = 5 21 RHS ⇒ b + a = 2 3 + - 3 7 = 2 3 - 3 7 = 14 - 9 21 = 5 21 Therefore, a + b = b + a Hence verified.

Verify the following: 6 7 + 0 = 0 + 6 7 = 6 7

Given to verify: 6 7 + 0 = 0 + 6 7 = 6 7 We know that the rational number 0 is the additive identity of any rational number, i.e. if a is a rational number, then a + 0 = 0 + a = a . LHS ⇒ 6 7 + 0 = 6 7 RHS ⇒ 0 + 6 7 = 6 7 Therefore, 6 7 + 0 = 0 + 6 7 = 6 7 Hence verified.