Question

5. Verify the property a× (b-c)= a×b-a×c for each of the following:
a= -3\5; b= 5\9; c= -10\3​

Answer 1

We conclude that

`a* \left(b-c\right)=\:a* b-a* c`

`-(7)/(3)=-(7)/(3)`

L.H.S = R.H.S

Explanation:

Given the property expression

`a* \left(b-c\right)=\:a* b-a* c`

Given that:

• a = -3/5

• b = 5/9

• c = -10/3

Determining the LEFT-HAND SIDE

`a* \left(b-c\right)`

substituting a= -3/5, b= 5/9 and c= -10/3​

`a* \left(b-c\right)\:=\:\:-(3)/(5)* \left((5)/(9)-\left(-(10)/(3)\right)\right)\:\:\:\:\:\:\:\:\:\:\:\:`

`=-(3)/(5)* \:\left((5)/(9)+(10)/(3)\right)\:\:\:`

`=-(3)/(5)* (35)/(9)`

`=-(7)/(3)`

Determining the RIGHT-HAND SIDE

`\:a* \:b-a* \:c`

substituting a= -3/5, b= 5/9 and c= -10/3​

`\:a* \:b-a* \:c=ab-ac`

`=-(3)/(5)\left((5)/(9)\right)-\left(-(3)/(5)\right)\left(-(10)/(3)\right)`

`=-(3)/(5)\cdot (5)/(9)-(3)/(5)\cdot (10)/(3)`

`=-(15)/(45)-(30)/(15)`

`=-(1)/(3)-2`

`=(-7)/(3)`

Apply the fraction rule:
`(-a)/(b)=-(a)/(b)`

`=-(7)/(3)`

Therefore, we conclude that

`a* \left(b-c\right)=\:a* b-a* c`

`-(7)/(3)=-(7)/(3)`

L.H.S = R.H.S