In this explainer, we will learn how to find square roots of perfect square integers, fractions, and decimals.
To understand square roots, we need to recall what squaring a number is.
Definition: Squaring a Number
Squaring a number consists in multiplying a number by itself.
We can think of the square of a number as the area of a square with that number for a side length.
For example, three squared (written ) is , and we can think of this as the area of the square with a side length of three.
Definition: Square Root
A square root of a number is a value that when multiplied by itself gives the number.
Unless otherwise stated, the square root of a number , written , will refer to the positive square root of that number.
For example, is defined as 3 and not , even though and .
How To: Taking the Square Root of a Number
The operation of taking the square root is the reverse of squaring a number. Taking the square root of a given number can be thought of as finding the side length of the square whose area is that number.
For instance, taking the square root of twenty-five (written ) means finding the side length of the square whose area is 25. We can see that it is 5, as illustrated in the diagram below.
In this explainer, we shall focus on finding the square roots of perfect squares.
Definition: Perfect Square
A perfect square is an integer that is the square of an integer.
So, for example, , , and are all perfect squares.
For any number that is a perfect square, it follows that must be an integer. This can be seen because we must have for some nonnegative integer , so taking the square roots of both sides gives . Looking at the right-hand side, since the operation of taking the square root is the reverse of squaring for nonnegative integers, then , which means that the value of is the integer .
With questions on this topic, it is important to pay careful attention to how they are expressed. In particular, the presence of the square root symbol in expressions of the form tells us to expect a single nonnegative answer; this is sometimes called the principal square root.
Let’s look at an example of this type.
Example 1: Finding Square Roots of Perfect Squares
Find the value of .
Answer
To find the value of , we need to consider a square of area 144. Trying out some examples of perfect squares, a square of side 10 has an area of , so this is too small. Similarly, a square of side 11 has an area of , which is also too small.
However, a square of side 12 does have an area of , as shown below.
We conclude that .
The above question wording featured a square root symbol, and this told us to expect a single nonnegative answer. Generally, however, every positive number has two square roots: and , which are sometimes written as . (This can easily be seen because just as the product of two positive numbers is positive, so is the product of two negative numbers: and .) So, if instead we had been asked to find the two square roots of 144, the correct answers would have been 12 and .
Now that we have learned how to find the square roots of integers that are perfect squares, we can extend these methods to find the square roots of fractions or decimals involving perfect squares. To do so, we need to introduce two important rules.
Rule: Product Rule
For nonnegative integers and , we have
The product rule allows us to transform the square root of a product into the product of two separate square roots. For example,
Similarly, the quotient rule, shown next, allows us to rewrite the square root of a fraction as the square root of the numerator divided by the square root of the denominator.
Rule: Quotient Rule
For positive integers and , we have
We are now in a position to tackle the next example, which involves a fraction (or rational number).
Example 2: Finding the Square Root of Rational Numbers
Find the two square roots of .
Answer
Finding the two square roots of the fraction is equivalent to finding
Henceforth, we will work with the positive square root; then, once we have evaluated it, we can just change the sign to get the negative one. Since we are dealing with the square root of a fraction, we can apply the quotient rule with and . In other words, this allows us to square root the numerator and denominator of the fraction separately, giving
Thus, we have reduced the problem to finding the values of and , before dividing the first by the second.
As and , then both 4 and 9 are perfect squares, with and . Therefore, we have shown that
To get the negative square root, we just change the signs in the above (which is equivalent to multiplying both sides of the equation by ), so we have . Thus, the two square roots of are and .
The above method can be applied to find the square roots of all nonnegative fractions (rational numbers) that have perfect square numerators and denominators. Our next example extends these ideas to decimals.
Example 3: Finding the Square Root of a Decimal Number
Evaluate .
Answer
In this question, we want to find the opposite (i.e., with an opposite sign) of the square root of 0.000169. It is very useful here to start by writing 0.000169 as a fraction. Remember that we get from 169 to 0.000169 by dividing by 1 000 000, so
Notice that and , so both the numerator and denominator of this fraction are perfect squares. Taking the square roots of both sides of the above, we have
The right-hand side features the square root of a fraction, so we can apply the quotient rule with and . In other words, this allows us to square root the numerator and denominator of the fraction separately, giving
Therefore, we have reduced the problem to finding the values of and , before dividing the first by the second.
Since , then . Likewise, implies . Thus, we have
As we were asked to find , we must multiply both sides of the equation by to obtain our final answer:
One advantage of the above method is that it enables us to find the square root of a decimal without having to use a calculator.
Our next example demonstrates how we can use similar techniques to find the square root of squared algebraic terms.
Example 4: Finding the Square Root of Squared Algebraic Terms
Simplify .
Answer
Here, we are asked to find the square root of an algebraic expression. Looking at the coefficient 100 and variable term separately, we notice that and . Thus, we deduce that the expression is a product of squares.
Next, it is important to note that the product rule can be applied to variable terms as well as numbers. This allows us to transform the square root of a product into the product of the two separate square roots. Therefore, in this case, we take and to get
Since , then . Also, implies . This means that we have shown that
Hence, the algebraic expression simplifies to .
As an interesting aside, in the example above, it was possible to apply the product rule to the term only because it is nonnegative for all values of . Similarly, the fact that implies followed from the fact that is nonnegative for all values of .
We can also use these ideas to solve related word problems. Here is an example taken from a geometric context where we will be able to find a length by taking the square root of a perfect square.
Example 5: Solving Word Problems Involving Square Roots in a Geometric Context
Given that and is the midpoint of , determine the length of .
Answer
Recall that taking the square root of a number can be thought of as finding the side length of the square whose area is that number.
The question tells us that the square of the length is equal to 100 cm2 and that is the midpoint of .
Our strategy will be to work out the length and then use this to calculate , which is the length of .
We know that
Taking the square roots of both sides, we get
On the left-hand side, the operation of taking the square root is the inverse of squaring, so simplifies to because lengths are nonnegative. Moreover, on the right-hand side, as , then 100 is a perfect square with . Therefore, the above equation simplifies to so we now know the length . As we are told that is the midpoint of , it must follow that , the length of , is half of the length . This value is .
Since the square of the length was given in square centimetres, it follows that any lengths must be in centimetres. We conclude that the length of is 5 cm.
Our last example is another word problem, and in this case, we will need to apply the product rule to obtain the solution.
Example 6: Solving Word Problems Involving Square Roots
A squared mosaic is made up of 1 800 white squares and 1 800 black squares of equal sizes. Determine the number of squares required to make one side of the mosaic.
Answer
Recall that we can think of taking the square root of a number as finding the side length of the square whose area is that number.
Here, we have a square mosaic made up of a number of smaller squares of equal sizes. If we calculate the total number of smaller squares, then finding the square root of this number will be equivalent to finding the number of squares required to make one side of the mosaic.
The total number of squares is
To determine the number of squares that make up one side of the mosaic, we need to work out , but notice first that . As and , then 3 600 is the product of two perfect squares. This means that we can apply the product rule with and to get
Since , then . Also , so . Therefore, we have shown that
We conclude that the number of squares required to make one side of the mosaic is .
Let’s finish by recapping some key concepts from this explainer.
Key Points
- Squaring a number consists in multiplying this number by itself.
- The operation of taking the square root is the reverse of squaring a number. We can think of taking the square root of a given number as finding the side length of the square whose area is that number.
- The square root symbol in an expression of the form denotes the positive square root of the number ; this is sometimes called the principal square root. Generally, however, every positive number has two square roots: and , which are sometimes written as .
- For any number that is a perfect square, it follows that both of its square roots must be integers.
- Product rule: for nonnegative integers and , we have .
- Quotient rule: for positive integers and , we have .
- To find the square root of a decimal without a calculator, it is helpful to write this decimal as a fraction and then apply the quotient rule.
- We can use the methods for finding the square roots of perfect square integers, fractions, and decimals to solve word problems, including those taken from a geometric context.
