Chapter 3, Problem 19CT

i.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $0=0+0$

Explanation of Solution

First take 0=0 then by adding 0 on the right hand side, we get$0=0+0$

Hence the property used here is identity property of addition

ii.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $a.0=a.\left(0+0\right)$

Explanation of Solution

From above part (i) we$0=0+0$ , now multiplying both sides by$a$ , we get$a.0=a.\left(0+0\right)$

here, by multiplying$a$ both sides we get this, so Multiplicative Property of equality is used

iii.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $a.0=a.0+a.0$

Explanation of Solution

From the above part (ii), we have$a.0=a.\left(0+0\right)$ , Now on the right hand side distributing$a$ over the addition, we get$a.0=a.0+a.0$

Hence here we use distributive property.

iv.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $a.0+a.0=a.0+0$

Explanation of Solution

We know that$a.0=0$

By using the property of part (i), we have$a.0=a.0+\mathrm{0...}\left(ii\right)$

Adding both above equations, we have

  $a.0+a.0=a.0+0$

here we substituted$a.0=a.0+0$ , so we used substitute property

v.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $a.0=0$

Explanation of Solution

From above part (iv) we have$a.0+a.0=a.0+0$ , Now subtracting$a.0$ both sides we get

  $a.0=0$ ,

Hence here subtraction property of equality is used

vi.

To determine

To explain: Write the missing reasons to justify the multiplicative property of zero.

  $0.a=0$

Explanation of Solution

From above part we have$a.0=0$ , by interchanging the position of 0 and$a$ , we get$0.a=0$ , so this implies$a.0=0.a$ , which is commutative property of multiplication