Could I please have help with describing mixed numbers (aka mixed fractions) that have this property:
Show that $\sqrt{9\frac{9}{80}}=9\sqrt{\frac{9}{80}}$ and $\sqrt{4\frac{4}{15}}=4\sqrt{\frac{4}{15}},\;$ where $\sqrt{9\frac{9}{80}} = \sqrt{9+\frac{9}{80}}$ etc.
I can easily show this mathematically but however, which mixed numbers have this property and which don't, is there a rule, etc.
HINT
For $x,y\in \mathbb{Z^{+}}$ $$\begin{align}\sqrt{x\dfrac{x}{y}} &= x\sqrt{\dfrac{x}{y}}\\ &\iff \\\sqrt{x+\dfrac{x}{y}}&=\sqrt{\dfrac{x^3}{y}}\\\sqrt{\dfrac{\color{blue}{x(y+1)}}{y}} &=\sqrt{\dfrac{\color{blue}{x^3}}{y}}\end{align}$$
Comparing you get $$\color{blue}{y = x^2-1}$$
That means it works for all $x,y$ that satisfy above equation
Check this for mixed numbers notation
Is not true, and neither is the other equality. $$\sqrt{9\frac{9}{80}}=\sqrt 9 \sqrt{\frac9{80}} = 3\sqrt{\frac{9}{80}}$$
– 5xum Mar 17 '15 at 08:03