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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.
The number 5 is a square root of 25, because $5^2 = 25$. The number $–5$ is another square root of 25, because $(–5)^2 = 25$.
The symbol $√a$ represents the nonnegative 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.)
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:
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 $$a/b$$:
For example, because 4 is a perfect square,
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
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.
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.
Complete this table for the equation $y=√{x^2}$.
This illustrates the following rule:
For all $x$, $√{x^2}=|x|$.
You can check that $|x|$ is a square root of $x^2$ algebraically:
Because $|x|$ is never negative, it must be the nonnegative square root of $x^2$ — that is to say, it must be $√{x^2}$.