How to determine significant figures involving radicals and exponents

Question

How do you determine the significant figures for solving equations with radicals and exponents? For example, how do you evaluate $x = \sqrt{4.56^2 +1.23^2}$?

Answer 1

Here is an abbreviation of the explanation I use in my 11th grade Chemistry and 12th grade Physics classes. This uses precision as is often used in American secondary schools, though it is not usually explained in quite this way.

There are two ways to measure precision: significant figures and decimal places. Significant figures (also called significant digits) are used in multiplication, division, powers, roots, and some other operations. Decimal places are used in addition and subtraction. In any operation, the proper precision of the answer equals the lowest precision of the operands. If multiple operations of the same kind are done consecutively, do rounding after doing all the operations. If two consecutive operations are of different kinds, you must round after each operation.

In your case, you have squaring, followed by addition, followed by a square root. This is significant figures followed by decimal places followed by significant figures, so we must round at each step.

First is squaring, which uses significant figures. If you think of this as powers, the $2$ is exact and does not affect the precision of the answer. In either case, each square has three significant figures, so we round each to three sig figs, getting $$x=\sqrt{20.8+1.51}$$

Next we add one decimal place to two decimal places, giving one decimal place. $$x=\sqrt{22.3}$$

Finally we take the square root of three significant figures, giving three sig figs. $$x=4.72$$

Is all that clear?