What are the possible solutions to 33 ⋅ 33 33 ⋅ 33 , if 11 ⋅ 11 = 4 11 ⋅ 11 = 4 and 22 ⋅ 22 = 16 22 ⋅ 22 = 16 ?

 What are the possible solutions to  33 ⋅ 33 33 ⋅ 33 , if  11 ⋅ 11 = 4 11 ⋅ 11 = 4  and  22 ⋅ 22 = 16 22 ⋅ 22 = 16 ?💝

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Let's analyze the problem by investigating the given equations and their possible patterns.

Given:

1. $11\cdot 11=4$
2. $22\cdot 22=16$

We need to find the solution to: $33\cdot 33$

First, let's determine if there's a pattern or a specific rule that applies to these operations.

Analyzing the Patterns

Let's denote the operation $\cdot$ as a function $𝑓$ such that: $𝑓(𝑎,𝑎)=𝑏$

Given:

1. $𝑓(11,11)=4$
2. $𝑓(22,22)=16$

Let's assume that $𝑓(𝑎,𝑎)$ might be dependent on the digits or some form of transformation of the numbers involved.

Checking for Digit-based Pattern

1. Consider $11\cdot 11=4$:

   • The digits of 11 are 1 and 1.
   • The sum of the digits: $1+1=2$.
   • Squaring the sum: $2^2=4$.

2. Consider $22\cdot 22=16$:

   • The digits of 22 are 2 and 2.
   • The sum of the digits: $2+2=4$.
   • Squaring the sum: $4^2=16$.

This pattern suggests that: $𝑓(𝑎,𝑎)={(\sum \text{digits of }𝑎)}^2$

Applying the Pattern to $33\cdot 33$

• The digits of 33 are 3 and 3.
• The sum of the digits: $3+3=6$.
• Squaring the sum: $6^2=36$.

Therefore, based on the identified pattern: $33\cdot 33=36$

Conclusion

The possible solution to $33\cdot 33$ is: $\boxed{{\displaystyle 36}}$