Calculating square roots might seem intimidating at first, but with the right approach, it becomes much more manageable! There are several different methods you can use to find square roots, and this article will walk you through them step-by-step so you can choose the method that works best for you.
Steps to Find a Square Root
To solve a square root, we'll follow the steps using this example:
1 If the radicand has more than two digits, we separate the digits into groups of two, starting from the right.
2 We calculate the integer or exact square root of the first group of digits on the left, in this case the (8)
We ask ourselves: what number squared gives 8?
We see that 8 is not a perfect square, but it's between two perfect squares: 4 and 9
So we take the square root of the perfect square that's less than (8), which is the square root of 4, giving us 2, and we place it in the corresponding box.
3 The square of the obtained root 2 (that is 4) is subtracted from the first group of digits that appear in the radicand (8)
In other words, the square of 2 is 4, we subtract it from 8 and get 4.
4 We bring down the next group of digits from the radicand, separating the first digit on the right (2) from the formed number (492) and dividing what remains by double the root 2, that is 2(2)=4.
In other words:
- We bring down 92, making the operable quantity of the radicand: 492.
- We separate the 1st digit on the right (2) and we're left with 49.
- We divide 49 by double the previously obtained root 2 · 2 = 4
- Since the result of 49 ÷ 4 is greater than 9, we take 9 as the result
Note: We take 9 whenever the division result (49÷4) is greater than 9
5 In another row below the root we place double the same (4). Next, we place the obtained quotient (9). Then the obtained number (49) is multiplied by said quotient (9). After that, we subtract (441) from the operable quantity (492) of the radicand.
- We place double the root in another row, which in this case is 4.
- We place the obtained quotient (9) next to the 4, thus obtaining the number 49.
- We multiply 49 by 9 and get 441.
- We subtract 441 from 492 (which is the operable quantity of the result).
Note: If we had obtained a value greater than the operable quantity of the radicand, we would have tried 8, then 7… until finding a smaller value.
Note: If the result of doing 49 · 9 had been greater than 492, we would have tried doing 49 · 8, 49 · 7,…
6 The obtained quotient (9) is the second digit of the root, leaving (29).
7 We bring down the next pair of digits and repeat the previous steps.
Since 5301 > 5124, we try 8.
Finally, we bring up the 8 to the root.
And with this we finish the process.
8 Square root verification. For the result to be correct, it must satisfy:
And indeed the obtained values satisfy it:
Square Root Examples
Solve the square roots of:
1 Calculate
Applying the previous process, we obtain:
finishing the process
2 Calculate
Let's see:
3 Calculate
Let's see:
What Is the Square Root of 169?
The square root of 169 is 13 – this is technically the principal square root of 169.
169=13
How to Find the Square Root of 169
When we approach the problem like this, we are finding the square root (sqrt) of the original number. The square root of x is a number, n, that satisfies this equation:
x=n2
Every positive number has two square roots, a positive and a negative.
The positive square root is called the principal square root, and it is generally understood to be the one we are interested in finding.
In this case, we want the positive square root of 169169.
Finding the Square Root of 169 by Factoring
To solve 169, we can factor the number under the radical sign. Prime factorization of the first several prime numbers will help us see which ones might be factors of the target number.
| Prime Number (n) | 169 / n | Does it factor? |
|---|---|---|
| 2 | 84.5 | No |
| 3 | 56.33 | No |
| 5 | 33.8 | No |
| 7 | 24.14286 | No |
| 11 | 15.36 | No |
| 13 | 13 | Yes |
| 17 | 9.941 | No |
Only one prime number is a factor, 13, and it times itself gives our target number, 169.
Rewrite the original problem, substituting our found factor for the original number:
169=13∗13=132
We cannot leave the exponent under the radical. To remove exponents from under radical signs, divide even exponents by 2, then move the base (in this case, 13) and its resulting exponent outside the radical sign.
Divide the exponent 2 by 2:
1322=131
Move 131 outside the radical sign and leave 1 under the radical sign (since 169 cannot be equal to nothing):
131=1311
This is the square root of 169 in radical form – this is the simplest radical form.
The new result looks complicated, but if you simplify the parts, you find it is not:
- 131=13
- 1=1
This yields 13 * 1, which is 13. The principal square root of 169 is 13.
The square root of 169 is a rational number because it is a perfect square -- the answer has no decimals.
How Do You Find the Square Root of 169 by Estimating?
Another way to find the square root of 169 is to estimate using known squares.
You probably instantly recall 122 is 144. Is 169 more or less than 144? It is more, so you need a bigger number.
Let's try 15. We calculate 152=225, which is too big.
We have now learned that the square root of 169 is somewhere between 12 and 15. Try the two remaining whole numbers.
142=196
Getting closer!
132=169
The square root of 169 is 13.
You can verify that you have the correct square root of a number by multiplying the number times itself to see if it equals the target number:
13∗13=169
Square Root of 169 by Division Method
Long division is another way to find a square root of 169 with a calculator.
Work from right to left and split the real number 169 into two pairs of two-digit numbers.
169=01ˉ69ˉ
Now, you work on each pair separately. What is the largest perfect square less than or equal to 1? The answer is 1. And the square root of 1 is also 1.
So, we put one on top and the bottom like this:
101ˉ69ˉ10ˉ
Now, subtract 1 from one and bring the remaining 69 down with your answer:
101ˉ69ˉ10ˉ69ˉ
The next step is to double the number at the very top, which is 1. So, 1 × 2 = 2.
Then, you use the number 2 and the remaining number at the bottom (69) to create this math problem:
2?×?≤69
Try and find which number fits to complete the equation. The largest number that works is 3. This means we have:
23×3=69
Now, you can add 3 to the top of your long division problem and 69 at the bottom; 69 minus 69 is 0, so you are done. The top is your square root of 169.
1301ˉ69ˉ10ˉ69ˉ069
The answer is 13.
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