Lesson Explainer: Square Roots of Perfect Squares Mathematics • 8th Grade

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 3๏Šจ) is 3ร—3, 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 ๐‘›, written โˆš๐‘›, will refer to the positive square root of that number.

For example, โˆš9 is defined as 3 and not โˆ’3, even though 3=9๏Šจ and (โˆ’3)=9๏Šจ.

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 โˆš25) 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, 1=1๏Šจ, 4=2๏Šจ, and 9=3๏Šจ are all perfect squares.

For any number ๐‘› that is a perfect square, it follows that โˆš๐‘› must be an integer. This can be seen because we must have ๐‘›=๐‘š๏Šจ for some nonnegative integer ๐‘š, so taking the square roots of both sides gives โˆš๐‘›=โˆš๐‘š๏Šจ. Looking at the right-hand side, since the operation of taking the square root is the reverse of squaring for nonnegative integers, then โˆš๐‘š=๐‘š๏Šจ, which means that the value of โˆš๐‘› is the integer ๐‘š.

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 ๏€บโˆš๏† in expressions of the form โˆš๐‘› tells us to expect a single nonnegative answer; this is sometimes called the principal square root.

Letโ€™s look at an example of this type.

Example 1: Finding Square Roots of Perfect Squares

Find the value of โˆš144.

Answer

To find the value of โˆš144, 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 10=10ร—10=100๏Šจ, so this is too small. Similarly, a square of side 11 has an area of 11=11ร—11=121๏Šจ, which is also too small.

However, a square of side 12 does have an area of 12=12ร—12=144๏Šจ, as shown below.

We conclude that โˆš144=12.

The above question wording featured a square root symbol, and this told us to expect a single nonnegative answer. Generally, however, every positive number ๐‘› has two square roots: โˆš๐‘› and โˆ’โˆš๐‘›, which are sometimes written as ยฑโˆš๐‘›. (This can easily be seen because just as the product of two positive numbers is positive, so is the product of two negative numbers: ๐‘›=โˆš๐‘›ร—โˆš๐‘› and ๐‘›=๏€บโˆ’โˆš๐‘›๏†ร—๏€บโˆ’โˆš๐‘›๏†.) So, if instead we had been asked to find the two square roots of 144, the correct answers would have been 12 and โˆ’12.

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 ๐‘Ž and ๐‘, we have โˆš๐‘Žร—๐‘=โˆš๐‘Žร—โˆš๐‘.

The product rule allows us to transform the square root of a product into the product of two separate square roots. For example, โˆš36=โˆš9ร—4=โˆš9ร—โˆš4=3ร—2=6.

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 ๐‘Ž and ๐‘, we have ๏„ž๐‘Ž๐‘=โˆš๐‘Žโˆš๐‘.

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 49.

Answer

Finding the two square roots of the fraction 49 is equivalent to finding ๏„ž49โˆ’๏„ž49.and

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 ๏„ž๐‘Ž๐‘=โˆš๐‘Žโˆš๐‘ with ๐‘Ž=4 and ๐‘=9. In other words, this allows us to square root the numerator and denominator of the fraction separately, giving ๏„ž49=โˆš4โˆš9.

Thus, we have reduced the problem to finding the values of โˆš4 and โˆš9, before dividing the first by the second.

As 2=2ร—2=4๏Šจ and 3=3ร—3=9๏Šจ, then both 4 and 9 are perfect squares, with โˆš4=2 and โˆš9=3. Therefore, we have shown that ๏„ž49=โˆš4โˆš9=23.

To get the negative square root, we just change the signs in the above (which is equivalent to multiplying both sides of the equation ๏„ž49=23 by โˆ’1), so we have โˆ’๏„ž49=โˆ’23. Thus, the two square roots of 49 are 23 and โˆ’23.

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 โˆ’โˆš0.000169.

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 1โ€Žโ€‰โ€Ž000โ€Žโ€‰โ€Ž000, so 0.000169=1691000000.

Notice that 169=13ร—13=13๏Šจ and 1000000=1000ร—1000=1000๏Šจ, so both the numerator and denominator of this fraction are perfect squares. Taking the square roots of both sides of the above, we have โˆš0.000169=๏„ž1691000000.

The right-hand side features the square root of a fraction, so we can apply the quotient rule ๏„ž๐‘Ž๐‘=โˆš๐‘Žโˆš๐‘ with ๐‘Ž=169 and ๐‘=1000000. In other words, this allows us to square root the numerator and denominator of the fraction separately, giving ๏„ž1691000000=โˆš169โˆš1000000.

Therefore, we have reduced the problem to finding the values of โˆš169 and โˆš1000000, before dividing the first by the second.

Since 13=169๏Šจ, then โˆš169=13. Likewise, 1000=1000000๏Šจ implies โˆš1000000=1000. Thus, we have โˆš0.000169=โˆš169โˆš1000000=131000=0.013.

As we were asked to find โˆ’โˆš0.000169, we must multiply both sides of the equation โˆš0.000169=0.013 by โˆ’1 to obtain our final answer: โˆ’โˆš0.000169=โˆ’0.013.

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 โˆš100๐‘ฅ๏Šง๏Šฌ.

Answer

Here, we are asked to find the square root of an algebraic expression. Looking at the coefficient 100 and variable term ๐‘ฅ๏Šง๏Šฌ separately, we notice that 10=10ร—10=100๏Šจ and ๏€น๐‘ฅ๏…=๐‘ฅร—๐‘ฅ=๐‘ฅ๏Šฎ๏Šจ๏Šฎ๏Šฎ๏Šง๏Šฌ. Thus, we deduce that the expression is a product of squares.

Next, it is important to note that the product rule โˆš๐‘Žร—๐‘=โˆš๐‘Žร—โˆš๐‘ 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 ๐‘Ž=100 and ๐‘=๐‘ฅ๏Šง๏Šฌ to get โˆš100๐‘ฅ=โˆš100ร—๐‘ฅ=โˆš100ร—โˆš๐‘ฅ.๏Šง๏Šฌ๏Šง๏Šฌ๏Šง๏Šฌ

Since 10=100๏Šจ, then โˆš100=10. Also, ๏€น๐‘ฅ๏…=๐‘ฅ๏Šฎ๏Šจ๏Šง๏Šฌ implies โˆš๐‘ฅ=๐‘ฅ๏Šง๏Šฌ๏Šฎ. This means that we have shown that โˆš100๐‘ฅ=10ร—๐‘ฅ=10๐‘ฅ.๏Šง๏Šฌ๏Šฎ๏Šฎ

Hence, the algebraic expression โˆš100๐‘ฅ๏Šง๏Šฌ simplifies to 10๐‘ฅ๏Šฎ.

As an interesting aside, in the example above, it was possible to apply the product rule to the term ๐‘ฅ๏Šง๏Šฌ only because it is nonnegative for all values of ๐‘ฅ. Similarly, the fact that ๏€น๐‘ฅ๏…=๐‘ฅ๏Šฎ๏Šจ๏Šง๏Šฌ implies โˆš๐‘ฅ=๐‘ฅ๏Šง๏Šฌ๏Šฎ followed from the fact that ๐‘ฅ๏Šฎ is nonnegative for all values of ๐‘ฅ.

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 (๐‘‹๐‘Œ)=100๏Šจ๏Šจcm and ๐‘ is the midpoint of ๐‘‹๐‘Œ, determine the length of ๐‘‹๐‘.

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 ๐‘‹๐‘Œ is equal to 100 cm2 and that ๐‘ is the midpoint of ๐‘‹๐‘Œ.

Our strategy will be to work out the length ๐‘‹๐‘Œ and then use this to calculate ๐‘‹๐‘, which is the length of ๐‘‹๐‘.

We know that (๐‘‹๐‘Œ)=100.๏Šจ

Taking the square roots of both sides, we get ๏„(๐‘‹๐‘Œ)=โˆš100.๏Šจ

On the left-hand side, the operation of taking the square root is the inverse of squaring, so ๏„(๐‘‹๐‘Œ)๏Šจ simplifies to ๐‘‹๐‘Œ because lengths are nonnegative. Moreover, on the right-hand side, as 10=10ร—10=100๏Šจ, then 100 is a perfect square with โˆš100=10. Therefore, the above equation simplifies to ๐‘‹๐‘Œ=10, so we now know the length ๐‘‹๐‘Œ. As we are told that ๐‘ is the midpoint of ๐‘‹๐‘Œ, it must follow that ๐‘‹๐‘, the length of ๐‘‹๐‘, is half of the length ๐‘‹๐‘Œ. This value is 10รท2=5.

Since the square of the length ๐‘‹๐‘Œ was given in square centimetres, it follows that any lengths must be in centimetres. We conclude that the length of ๐‘‹๐‘ 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 1โ€Žโ€‰โ€Ž800 white squares and 1โ€Žโ€‰โ€Ž800 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 1800+1800=3600.

To determine the number of squares that make up one side of the mosaic, we need to work out โˆš3600, but notice first that 3600=36ร—100. As 6=6ร—6=36๏Šจ and 10=10ร—10=100๏Šจ, then 3โ€Žโ€‰โ€Ž600 is the product of two perfect squares. This means that we can apply the product rule โˆš๐‘Žร—๐‘=โˆš๐‘Žร—โˆš๐‘ with ๐‘Ž=36 and ๐‘=100 to get โˆš3600=โˆš36ร—100=โˆš36ร—โˆš100.

Since 6=36๏Šจ, then โˆš36=6. Also 10=100๏Šจ, so โˆš100=10. Therefore, we have shown that โˆš3600=6ร—10=60.

We conclude that the number of squares required to make one side of the mosaic is โˆš3600=60.

Letโ€™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 โˆš in an expression of the form โˆš๐‘› denotes the positive square root of the number ๐‘›; this is sometimes called the principal square root. Generally, however, every positive number ๐‘› has two square roots: โˆš๐‘› and โˆ’โˆš๐‘›, which are sometimes written as ยฑโˆš๐‘›.
  • For any number that is a perfect square, it follows that both of its square roots must be integers.
  • Product rule: for nonnegative integers ๐‘Ž and ๐‘, we have โˆš๐‘Žร—๐‘=โˆš๐‘Žร—โˆš๐‘.
  • Quotient rule: for positive integers ๐‘Ž and ๐‘, we have ๏„ž๐‘Ž๐‘=โˆš๐‘Žโˆš๐‘.
  • 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.

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