Question

Let n(X*Y)=24, n(X*Z)=15, and n(Y*Z)=40. Find n(X*Y*Z).

Equation, expression, and inequalitie
ANSWERED
asked 2021-06-13
Let \(\displaystyle{n}{\left({X}\cdot{Y}\right)}={24},{n}{\left({X}\cdot{Z}\right)}={15}\), and \(\displaystyle{n}{\left({Y}\cdot{Z}\right)}={40}\). Find n(X*Y*Z).

Answers (1)

2021-06-14

We know that: \(\displaystyle{n}{\left({X}\cdot{Y}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\), \(\displaystyle{n}{\left({X}\cdot{Z}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Z}\right)}\), \(\displaystyle{n}{\left({Y}\cdot{Z}\right)}={n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\),
and
\(\displaystyle{x}{\left({X}\cdot{Y}\cdot{Z}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\)
Therefore, \(\displaystyle{n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}={24},{n}{\left({X}\right)}\cdot{n}{\left({Z}\right)}={15}\)
and
\(\displaystyle{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}={40}\)
Multiplying the three equations, we get: \(\displaystyle{\left({n}{\left({X}\right)}\right)}^{{2}}{\left({n}{\left({Y}\right)}\right)}^{{2}}{\left({n}{\left({Z}\right)}\right)}^{{2}}={24}\cdot{15}\cdot{40}={14400}\)
Thus, \(\displaystyle{\left({n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\right)}^{{2}}={14400}\to{n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}={120}\)
Finally, using (*), \(\displaystyle{n}{\left({X}\cdot{Y}\cdot{Z}\right)}={120}\)

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