# How to determine significant figures involving radicals and exponents

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

• Please see meta.math.stackexchange.com/questions/5020/… for information about how to write mathematics on this site. – N. F. Taussig Jan 26 '15 at 15:18
• Are you asking about the number of significant figures in the answer or how to do the calculations to arrive at the answer? – N. F. Taussig Jan 26 '15 at 15:22

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?

• What would be the scenario on handling powers? i.e. $5.1^4$ – Another.Chemist Jan 27 '17 at 19:21
• @Another.Chemist: This is like squaring: powers use significant digits. The $4$ is (probably) exact, so we ignore that for deciding precision, so the answer should have two significant digits, just as $5.1$ does. That give an answer of $680$. Note that the trailing zero when there is no decimal point in the number is not significant. This method has its weaknesses, since $5.05^4$ is about $650$ and $5.15^4$ is about $703$. I teach my classes that this method is simple and useful but has its problems. My classes like simple. – Rory Daulton Jun 15 '17 at 18:19
• @RoryDaulton I do not understand this sentence: "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". Why not always rounding after all the operations? – Emmet Oct 23 '17 at 16:40
• @Emmet: If you try to round only after all the operations (with a mixture of significant-digit and decimal-point operations) it is difficult to know which precision to use for your final answer. Rounding after each operation makes that easier. – Rory Daulton Oct 23 '17 at 22:03
• @RoryDaulton I see, thank you. – Emmet Oct 24 '17 at 7:33