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Approximating Square Roots

When solving a problem with square roots, life will not always give you perfect squares. We suggest that you memorize the first $12$ perfect squares. For the most part we won't work with numbers bigger than $12^2$.

That way, you can easily determine if a number under a root sign is a perfect square! Remember, this sign looks like

$$\sqrt{}$$

There will be a nonnegative real number underneath, usually a whole number.

Remember that perfect squares are produced by multiplying positive numbers by themselves. For instance, multiply the number $5$ by itself to get $5 \times 5 = 5^2 = 25$.

The perfect squares are as follows: $$\begin{align} 1^2 &= 1\\ 2^2 &= 4\\ 3^2 &= 9\\ 4^2 &= 16\\ 5^2 &= 25\\ 6^2 &= 36\\ 7^2 &= 49\\ 8^2 &= 64\\ 9^2 &= 81\\ 10^2 &= 100\\ 11^2 &= 121\\ 12^2 &= 144\\ \end{align}$$

Alongside that, here's a list of the square roots of perfect squares.

$$\begin{align} \sqrt{1} &= 1 \\ \sqrt{4} &= 2 \\ \sqrt{9} &= 3 \\ \sqrt{16} &= 4\\ \sqrt{25} &= 5\\ \sqrt{36} &= 6\\ \sqrt{49} &= 7\\ \sqrt{64} &= 8\\ \sqrt{81} &= 9\\ \sqrt{100} &= 10\\ \sqrt{121} &= 11\\ \sqrt{144} &= 12 \end{align}$$

How to Handle Non-perfect Squares

Now, there is a secret to solving non-perfect squares. That is, sometimes we want to calculate square roots of numbers that are not perfect squares. There's an equation we can use to approximate a square root! It is: $$\sqrt{c} = \sqrt{a + b} \approx \sqrt{a} + \frac{b}{2\sqrt{a} + 1}$$

In this equation, $c$ is the number for which we want to guess $\sqrt{c}$. It may be called the non-perfect square. Notice that $a$ is the first perfect square that is smaller than $c$. Furthermore, $b$ is the difference between $c$ and $a$; in other words, $b = c - a$. Here, $c$ will always be given in the problem. Also, $a$ will be on the list of perfect squares. Therefore, we can always find $b$.

Make sure to refresh yourself on topics such as order of operations, variables or fractions if needed! These topics will help in solving the problem. That's because we will be solving for variables and plugging them into this equation.

Example

Let's solve a non-perfect square together!

What is $\sqrt{10}$? We want to find a number that is close to the value of $\sqrt{10}$.

Immediately, we notice that $c = 10$. This is because $10$ is the only non-perfect square in the problem. Now, we look at the list of perfect squares. The largest perfect square that is less than $10$ is $9$. Therefore, $a = 9$. Lastly, let's find $b$. We know that $b = c - a$, so we have $b = 10 - 9$. That means $b = 1$.

It is important for us to solve for our three variables before we do anything else. That way, we can plug all of the corresponding numbers into the equation, which makes this problem a lot easier!

$$\begin{align} \sqrt{c} = \sqrt{a + b} &= \sqrt{a} + \frac{b}{2\sqrt{a} + 1}\\ \\ \sqrt{10} = \sqrt{9 + 1} &= \sqrt{9} + \frac{1}{2\sqrt{9} + 1}\\ &= 3 + \frac{1}{2(3) + 1}\\ &= 3 + \frac{1}{6 + 1}\\ &= 3 + \frac{1}{7}\\ \end{align}$$

Our final answer is the mixed number $3 \frac{1}{7}$. It can also be written as the decimal $3.143\dots$ or the improper fraction $\frac{22}{7}$.

Now, does this answer make sense? Let's double check our work.

Like we mentioned above, $9$ is the first perfect square that appears below $10$. If we look for the first perfect square that appears above $10$, we notice that it is $16$. Here's that list of perfect squares again as a reference.

$$\begin{align} 1^2 &= 1\\ 2^2 &= 4\\ 3^2 &= 9\\ 4^2 &= 16\\ 5^2 &= 25\\ 6^2 &= 36\\ 7^2 &= 49\\ 8^2 &= 64\\ 9^2 &= 81\\ 10^2 &= 100\\ 11^2 &= 121\\ 12^2 &= 144\\ \end{align}$$

Therefore, we can write a pair of inequalities... $$\sqrt{9} < \sqrt{10} < \sqrt{16}$$ This can be reduced to: $$3 < \sqrt{10} < 4$$ We can reduce this because $9$ and $16$ are on the list perfect squares. The numbers on the ends will always be perfect squares. We like this because after taking a square root, the ends will still be whole numbers.

Comparing this to our final answer of $3.143$, we can see that: $$3 < 3.143 < 4$$ This is a true statement, so we know that our answer is valid! Let's check to see if squaring $3.143$ will give us something close to $10$. Remember, $3.143$ is the decimal form of our approximation.

$$3.143^2 = 9.88$$

Note: The processes used in this section is used for APPROXIMATING square roots. This means that these are NOT exact answers, but they get us close! Since $3.143^2$ is close to $10$, that means our formula gives a good guess. Finally, we can use a similar technique to handle cube roots.

Can I use a calculator?

Nowadays many calculators are programmed with buttons to do square roots. Better calculators will give you more digits, and therefore a more accurate answer. However, no calculator can show all the digits.

Why's that?

It's because irrational numbers are decimals that do not terminate or repeat. The decimal expansions for irrational numbers go on forever. However, finding a large number of digits is as easy as pushing a few buttons on a calculator. That sounds like a lot less work than the approximation technique we just described. However, the other technique gives a better idea of how to visualize comparing numbers to each other.

Square Roots vs. Exponents

Despite what you may think, square roots and exponents are closely related. If we square a number, we are multiplying that number by itself. For instance,

$$3^2 = 3 \times 3 = 9$$

Think of a square root as splitting the number on the right into the product of a different number with itself. That number you split into is the square root:

$$\sqrt{9} = \sqrt{3 \times 3} = 3$$

As a result we call squaring and square roots inverse operations of each other. If we square a number, and then take its square root, we get the number we started with. A perfect square is a number whose square root is a positive integer ($1, 2, 3, \dots$). All other positive integers are called imperfect squares, and their square roots are irrational. Their square roots are harder to find exactly. Thus we developed ways to estimate these square roots.

Estimating Square Roots

Here is a three-step process for approximating square roots. This process is useful if the number we start with is NOT a perfect square.

Step One

Identify the next highest and lowest perfect squares around the number you are taking the square root of. For example, say you are trying to find $\sqrt{27}.$ The next highest and lowest perfect squares around $27$ are $36$ and $25.$ Notice that $27$ is closer to $25$ than it is to $36.$ This can be written with inequality notation:

$$25 < 27 < 36$$

This notation reads: $25$ is less than $27$, which is less than $36$.

Step Two

Take a root of the entire inequality and simplify. This is the same as taking the $\frac{1}{2}$ power of the entire inequality.

$$25 < 27 < 36$$ $$\Rightarrow \sqrt{25} < \sqrt{27} < \sqrt{36}$$ $$\Rightarrow 5 < \sqrt{27} < 6$$

The first and last numbers are simplified, but the one in the middle is not. If the middle one could be simplified, we wouldn't need this technique.

Step Three

Approximate! Since $27$ is closer to $25$ than it is to $36,$ $\sqrt{27}$ is closer to $\sqrt{25}=5$ than it is to $\sqrt{36}=6.$ Thus, we can approximate: $$\sqrt{27} \approx 5.1$$

We choose $5.1$ as our approximation as it is much closer to $5$ than it is to $6$. You could see this by looking at all three numbers on a number line.

This is a pretty good approximation! If we use a calculator, we can see that $\sqrt{27} = 5.19615242271\dots$ Then, if we round to two places past the decimal point, we have $\sqrt{27} \approx 5.20$.

Let's do another example! You'll want to repeat steps from the picture above.

Step One:

Identify the next highest and lowest perfect squares around $73.$ After some thought, we can determine that these are $64$ and $81.$ Notice that $73$ is in the middle-ish between $64$ and $81.$

Step Two:

Calculate square roots of all three numbers and simplify. $$64 < 73 < 81$$ $$\Rightarrow \sqrt{64} < \sqrt{73} < \sqrt{81}$$ $$\Rightarrow 8 < \sqrt{73} < 9$$

For right now we can only simplify the left and right-hand ends. They simplify to $8$ and $9$ because $8^2 = 64$ and $9^2 = 81$.

Step Three

Approximate! Since $73$ is halfway-ish between $64$ and $81, \sqrt{73}$ is halfway-ish between $\sqrt{64}=8$ and $\sqrt{81}=9.$ Thus, we can approximate: $$\sqrt{73} \approx 8.5$$

We use this guess because $8.5$ is halfway between $8$ and $9$.

This is a pretty good approximation! If we use a calculator, we can see that $\sqrt{73} = 8.54400374532\dots$

(Video) Square Root of any Number Instantly

by tecmath

In the above video by tecmath, we see a quick and easy way of finding square roots of non-perfect squares.

$\large{\text{Summary}}$

Disclaimer: This technique requires you to have a working knowledge of perfect squares.

Disclaimer: This is fairly accurate, but not $100\%$ accurate.

Suppose you want to find $\sqrt{87}$. This isn't a perfect square, so it won't be easy. Luckily, there's a trick. We will approximate $\sqrt{87}$ with a mixed number.

1. Find the largest perfect square that is below our number. In our example, this would be $81$, and $\sqrt{81}=9$. The square root of this lower-value perfect square will be the whole number part of our final mixed number.
2. Find the difference between the original number and the perfect square. In our example, $87-81=6$. This will be the numerator in our mixed number's fraction part.
3. Take the integer and double it. In our example, $9\times2=18$. This will be the denominator in our mixed number's fraction part.

So following these steps we get: $$9\frac6{18}=9\frac13\approx9.33$$

So, how accurate were we? Solving with a calculator: $$\sqrt{87}=9.327\dots$$ If we round to the nearest hundredth, we get the same number!

Following the steps above:

1. The first perfect square under $39$ is $36$. $\sqrt{36} = 6$
2. $39 - 36 = 3$
3. $6 \times 2 = 12$

After following these steps, our final mixed numbers is $6 \frac{3}{12}$. This can be reduced to $6 \frac{1}{4}$, which in decimal form is $6.25$