Problem: If abc = 12 and a²b²c² = 64, find the value of abbcac.

Solution:

To solve the problem, we need to use the given information to express abbcac in terms of abc. We can do this by simplifying a²b²c² and then substituting abc for 12 in the resulting expression.

Simplifying a²b²c²:

a²b²c² = (abc)² (product of powers property)
= (12)²
= 144

Substituting abc for 12:

abbcac = (ab)(bc)(ca)
= (abc)²
= 12²
= 144

Therefore, the value of abbcac is 144.

Explanation:

The problem involves finding the value of abbcac given two equations: abc = 12 and a²b²c² = 64. To solve the problem, we need to use algebraic manipulation to express abbcac in terms of abc, which we can then substitute with the given value of 12.

We start by simplifying a²b²c² using the product of powers property. This gives us a simpler expression that contains only abc. We then substitute abc for 12 in the resulting expression to get the value of abbcac.

The solution shows that the value of abbcac is 144. This means that the product of ab, bc, and ca is 144 given the constraints of the problem.

Conclusion:

The value of abbcac is 144. To arrive at this answer, we simplified a²b²c² as (abc)² and then substituted abc for 12. The resulting expression gave us the value of abbcac, which is equal to 144.

(a+b+c)²=a²+b²+c²+2ab+2bc+2ac=12²
=64+2(ab+bc+ac)=144
=2(ab+bc+ca)=144-64=80
=ab+bc+ca=80/2
=40