To solve the problem, we need to find the value of G41+G42+G43+G21G23 where G1, G2, G3 are the geometric means of two distinct positive numbers A and B.
1. Define the Numbers:
Let the two distinct positive numbers be A and B.
2. Find the Geometric Means:
The geometric means G1, G2, G3 can be defined as follows:
3. Calculate G41, G42, and G43:
4. Sum the Fourth Powers:
Now we sum these results: G41+G42+G43=AB3+A2B2+A3B
5. Calculate G21G23:
Now, we need to calculate G21G23:
6. Combine All Terms:
Now we combine all the terms: G41+G42+G43+G21G23=(AB3+A2B2+A3B)+A2B2
This simplifies to: AB3+2A2B2+A3B
7. Factor the Expression:
We can factor this expression: =AB(A2+2AB+B2)=AB(A+B)2
Final Answer:
Thus, the expression G41+G42+G43+G21G23 is equal to: AB(A+B)2
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