I noticed an anomaly when using different ways to calculate square roots. For good or bad over the years I have fallen into a habit of calculating square roots as raising a number to .5. Doing this in Mathematica has caused an unexplained behavior.

If I use Sqrt[5] or (5^(1/2) I get the same results and displayed precision If I use (5^.5) I get the same result but less displayed precision.

See the below example . For phi1 and phi 2 I can see 20 decimal places. For phi3 I can see at most 5 decimal places. However phi1-phi3 is zero so I am getting the same result

Why am I only seeing 5 decimal places for phi3?

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$Version

phi1 = (1 + Sqrt[5])/2;
N[phi1, 20]                     (* <——— Shows 20 decimal places*)

phi2 = (1 + 5^(1/2))/2;
N[phi2, 20]                    (* <——— Shows 20 decimal places*)

phi3 = (1 + 5^(.5))/2;
N[phi3, 20]                    (* <——— Shows only 5 decimal places*)

phi1-phi3


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"11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)"

1.6180339887498948482

1.6180339887498948482

1.61803

0.