Square Roots

When a number is raised to the second power, we say that the number is squared. For example, $4^2=4(4)$ is referred to as “4 squared.” Often we need to know what number was squared in order to produce some value $a$. If we can find such a number, we call that number a square root of $a$. In this lesson you will learn about how to take square roots, and some of their properties.


Defining and computing square roots

The number 5 is a square root of 25, because $5^2 = 25$. The number $–5$ is another square root of 25, because $(–5)^2 = 25$.

Find the two square roots of 4.
Find the square roots of 36.
Is the square of a positive number positive, negative, or zero? (For example, is $3^2=3(3)$ positive, negative, or zero?)
Is the square of a negative number positive, negative, or zero? (For example, is $(-4)^2=(-4)(-4)$ positive, negative, or zero?)
Is $0^2$ positive, negative, or zero?
Can the square of a real number be negative?
Is it possible for a negative number to have a real square root?

The symbol $√a$ represents the non-negative square root of $a$. For example, $√25 = 5$ and $–√25 = –5$. We can illustrate $√a$ on the grid to the left. For example, at the moment, the grid shows a square with area 25. Because the sides of that square have length 5, this shows that $√25=5$.

Find the square roots in the table below.

All the numbers whose square roots you have been asked about so far are perfect squares: that is, each is a whole number that is the square of a whole number. If you are asked to find a square root of a number which is not a perfect square, you should find it by using a calculator, and write it as a decimal.

Using a calculator, find the square roots in the table below, rounded to four decimal places. (We write $≈$ — meaning “approximately equal to” — as a reminder that your answer is rounded.)

Products and quotients of square roots

Compute the quantities shown in the table below, rounded to four decimal places. Use a calculator as necessary.

As you have seen:

Whenever $a ≥ 0$ and $b ≥ 0$, $√a√b = √{ab}$.

You can check algebraically that $√a√b$ is a square root of $ab$, using the associative and commutative properties of multiplication:

$$ \cl"tight"{\table , (√a√b)^2; =, (√a√b)(√a√b); =, √a√b√a√b; =, √a√a√b√b; =, (√a√a)(√b√b); =, ab; } $$

Compute the quantities shown in the table, to four decimal places.

Similarly to the last question, you can see that:

Whenever $a ≥ 0$ and $b > 0$, $$√a/√b = √{a/b}$$.

You can verify algebraically that $$√a/√b$$ is a square root of $$a/b$$:

$$ \cl"tight"{\table , (√a/√b)^2; =, (√a/√b)(√a/√b); =, {√a√a}/{√b√b}; =, a/b; } $$

For example, because 4 is a perfect square,

$$√{4x}=√4√x=2√x$$

for all $x ≥ 0$, as shown on the grid to the left. This can be used to simplify the square roots of numbers that are multiples of 4.

Simplify the square roots in the table below by using the fact that $√{4x}=2√x$.

You can check that the result of this simplification is actually a square root of the original number. For example, we just found that $√12=2√3$; you can check this by noticing that

$$ (2√3)^2=2√3(2√3)=2(2)√3√3=4(3)=12 $$ meaning that $2√3$ actually is a square root of 12.

Simplify the square roots in the table below, by removing the largest possible perfect square from the square root. Then check that the simplified form you get is actually a square root of the original number.

Similarly, you can often simplify fractions inside square roots by using the division rule.

Using the fact that $$√{a/b}={√a}/{√b}$$ whenever $a ≥ 0$ and $b > 0$, simplify these fractions and check your simplification.

Addition and subtraction with square roots

In the last section you learned that, when you’re working only with positive numbers, you can do multiplication and division either before or after you take square roots, and you’ll end up with the same result. In this question, we’ll look at whether this also works for addition and subtraction.

Compute the quantities to the right, as pictured on the grid to the left.
Is $√{16+9}=√16+√9$?
Could there be a general rule saying that $√{a+b}=√a+√b$?
Click . Compute the quantities to the right, which are now pictured on the grids to the left.
Is $√{100-64}=√100-√64$?
Could there be a general rule saying that $√{a-b}=√a-√b$?

Square roots and absolute values

Complete this table for the equation $y=√{x^2}$.

$x$$x^2$$y=√{x^2}$
Click to see the complete graph of $y=√{x^2}$. Which letter of the alphabet is this graph shaped like?
The other graph we’ve seen with this shape is the graph of $y=|x|$. Click to compare these two graphs. Are they the same or are they different?

This illustrates the following rule:

For all $x$, $√{x^2}=|x|$.

You can check that $|x|$ is a square root of $x^2$ algebraically:

$$ \cl"tight"{\table {|x|^2}, =, {|x||x|}; , =, {|x(x)|}, {\; \text"(because "\, |a||b|=|ab| \text")"}; , =, {|x^2|}; , =, {x^2}, {\; \text"(because "\, x^2 ≥ 0 \, \text"always)"} } $$

Because $|x|$ is never negative, it must be the non-negative square root of $x^2$ — that is to say, it must be $√{x^2}$.