Calculating the Square Root of 27: How-To & Steps

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

27 isn't a perfect square, so how do we find its root? In this lesson, we will understand an iterative method that uses the basic operations of addition, subtraction, multiplication and division for calculating the square root of 27 and any other number.

Find the Square Root X

Step 1: Decide on the desired accuracy of the answer.

The method is iterative and will lead to an answer which is as accurate as we want. We base the accuracy on how close the square of our answer is to X.

Let's choose an accuracy of .01%

Step 2: Calculate the range of acceptable answers.

Knowing the lowest and the highest acceptable answers makes it easier to decide when we have a good answer and can stop the method.

.01% of 27 is .0001(27) = .00027.

We are looking for a square root of 27 which when squared will be between 27 - .00027 and 27 + .00027. Doing the subtraction and addition, we get a range between

26.9973 and 27.0027.

Step 3: Make a guess, g, at the square root of X

The iterative method starts with a guess, g, for the square root of X.

27 is close to 25 which has a square root of 5. We will choose g = 5 but it turns out any number can be used as the first guess.

Step 4: Compute g2

We simply calculate the square of g.

g2 = 52 = 25.

Step 5: Stop or continue?

If g2 is within the range of acceptable answers, stop. The answer is g.

Otherwise, continue by replacing g with the average of g and X/g:


replace_g


This left-pointing arrow means we substitute the current values of X and g on the right-hand side and calculate a number. We take this number as the new g. The arrow tells us to calculate the right-hand side and assign this as the new value for g on the left-hand side. If g were the exact answer, then g and X/g would be the same. The average would be just g. However, if g is not yet the answer, taking the average of g with X/g gets us closer to the answer. After updating g, we go back to STEP 4.

We keep doing this loop of STEP 4 and STEP 5 until we get a g2 which is within the acceptable range.

Getting back to our method, we squared the guess of 5 and got 25. However, 25 is not within the 26.9973 and 27.0027 so we continue by replacing g with the average of g and X/g:


replace_g


The details of calculating the right-hand side:


rhs_details


The right-hand side computes to 5.2 and this becomes the new value for g.

Going back to Step 4, compute g2 = 5.22 = 27.04.

We are still outside the range of 26.9973 to 27.0027, so again we update g:


replace_g


Calculating the right-hand side:


rhs


The new g computes to 5.19615… and g2 = 26.99997… which is in the desired range.

Thus, we stop and the answer is 5.19615…

In fact, if we round this answer to 4 significant figures, we are still within the desired range. Meaning, 5.19615… rounded to 4 significant figures is 5.196 and 5.1962 = 26.9984… which is greater than 26.9973 but less than 27.0027. By the way, 3 significant figures or less for this g will not satisfy the desired accuracy.

The Final Answer for the Square Root of 27

To an accuracy of at least .01%, the square root of 27 is 5.196.

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