This simplifies to:

2 cdot 3 5

6 5

This equation is clearly false, as 6 does not equal 5. Therefore, the initial assumption that sqrt{2} cdot sqrt{3} sqrt{5} must be incorrect. Consequently, we can conclude:

sqrt{2} cdot sqrt{3} eq sqrt{5}

Numerical Approximation for Clarity

For further clarity, let’s evaluate the numerical values:

Calculating the product:

sqrt{2} cdot sqrt{3} approx 1.414 cdot 1.732 approx 2.449

Clearly, 2.449 eq sqrt{5} approx 2.236. Hence, mathematically, sqrt{2} cdot sqrt{3} does not equal sqrt{5}.

Further Clarification with Algebraic Steps

Let’s verify the initial assumption by using the fact that if a b, then a^2 b^2. Therefore, if:

sqrt{2} cdot sqrt{3} sqrt{5}

then:

(sqrt{2} cdot sqrt{3})^2 (sqrt{5})^2

Squaring both sides gives:

2 cdot 3 5

6 5

This equation is true only if sqrt{6} 0, which is false. Consequently, the assumption that sqrt{2} cdot sqrt{3} sqrt{5} is incorrect.

Conclusion

The product of sqrt{2} cdot sqrt{3} is always sqrt{6}. Since sqrt{6} eq sqrt{5}, the statement sqrt{2} cdot sqrt{3} sqrt{5} is false. Therefore, we can confidently conclude:

sqrt{2} cdot sqrt{3} eq sqrt{5}