Lösung 3.1:8d
Aus Online Mathematik Brückenkurs 1
In power form, the expressions become
Admittedly, it is true that \displaystyle 2^{1/2} > 2^{1/3} and \displaystyle 3^1 > 3^{3/4}, but this does not help us to say anything about how the products are related to each other. Instead, we observe that the exponents 1/2, 3/4, 1/3 and 1 have \displaystyle 3\cdot 4 = 12 as the lowest common denominator which we can take out
Now, we can compare the bases \displaystyle 2^6\cdot 3^9 and \displaystyle 2^4\cdot 3^{12} with each other and so decide which number is larger.
Because
the denominator \displaystyle 2^{4}\cdot 3^{12} is larger than the numerator \displaystyle 2^6\cdot 3^9, which means that \displaystyle \sqrt[3]{2}\cdot 3 is larger than \displaystyle \sqrt{2}\bigl(\sqrt[4]{3}\bigr)^{3}.
