The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 0 is 0, the square root of 100 is 10 and the square root of 50 is 7.071. Sometimes, you can figure out, or simply recall, the square root of a number that itself is a "perfect square," which is the product of an integer multiplied by itself; as you progress through your studies, you're likely to develop a mental list of these numbers (1, 4, 9, 25, 36 . . .).
Problems involving square roots are indispensable in engineering, calculus and virtually every realm of the modern world. Although you can easily locate square root equation calculators online (see Resources for an example), solving square root equations is an important skill in algebra, because it allows you to become familiar with using radicals and work with a number of problem types outside the realm of square roots per se.
Squares and Square Roots: Basic Properties
The fact that multiplying two negative numbers together yields a positive number is important in the world of square roots because it implies that positive numbers actually have two square roots (for example, the square roots of 16 are 4 and −4, even if only the former is intuitive). Similarly, negative numbers do not have real square roots, because there is no real number that takes on a negative value when multiplied by itself. In this presentation, the negative square root of a positive number will be ignored, so that "square root of 361" can be taken as "19" rather than " −19 and 19."
Also, when trying to estimate the value of a square root when no calculator is handy, it is important to realize that functions involving squares and square roots are not linear. You'll see more on this in the section about graphs later, but as a rough example, you have already observed that the square root of 100 is 10 and the square root of 0 is 0. On sight, this might lead you to guess that the square root for 50 (which is halfway between 0 and 100) must be 5 (which is halfway between 0 and 10). But you have also already learned that the square root of 50 is 7.071.
Finally, you may have internalized the idea that multiplying two numbers together yields a number greater than itself, implying that square roots of numbers are always smaller than the original number. This is not the case! Numbers between 0 and 1 have square roots, too, and in every case, the square root is greater than the original number. This is most easily shown using fractions. For example, 16/25, or 0.64, has a perfect square in both the numerator and the denominator. This means that the square root of the fraction is the square root of its top and bottom components, which is 4/5. This is equal to 0.80, a greater number than 0.64.
Square Root Terminology
"The square root of x" is usually written using what is called a radical sign, or just a radical (√ ). Thus for any x:
represents its square root. Flipping this around, the square of a number x is written using an exponent of 2 (x2). Exponents take superscripts on word-processing and related applications, and are also called powers. Because radical signs are not always easy to produce on demand, another way to write "the square root of x" is to use an exponent:
This in turn is part of a general scheme:
means "raise x to the power of y, then take the 'z' root of it." x1/2 thus means "raise x to the first power, which is simply x again, and then take the 2 root of it, or the square root." Extending this, x(5/3) means "raise x to the power of 5, then find the third root (or cube root) of the result."
Radicals can be used to represent roots other than 2, the square root. This is done by simply appending a superscript to the upper left of the radical.
then, represents the same number as x(5/3) from the previous paragraph does.
Most square roots are irrational numbers. This means that not only are they not nice, neat integers (e.g., 1, 2, 3, 4 . . .), but they also cannot be expressed as a neat decimal number that terminates without having to be rounded off. A rational number can be expressed as a fraction. So even though 2.75 is not an integer, it is a rational number because it is the same thing as the fraction 11/4. You were told earlier that the square root of 50 is 7.071, but this is actually rounded off from an infinite number of decimal places. The exact value of √50 is 5√2, and you'll see how this is determined soon.
Graphs of Square Root Functions
You have already seen that equations in involving squares and square roots are nonlinear. One easy way to remember this is that the graphs of the solutions of these equations are not lines. This makes sense, because if, as noted, the square of 0 is 0 and the square of 10 is 100 but the square of 5 is not 50, the graph resulting from simply squaring a number must curve its way to the correct values.
This is the case with the graph of
as you can see for yourself by visiting the calculator in the Resources and changing the parameters. The line passes through the point (0,0), and y does not go below 0, which you should expect because you know that x2 is never negative. You can also see that the graph is symmetrical around the y-axis, which also makes sense because every positive square root of a given number is accompanied by a negative square root of equal magnitude. Therefore, with the exception of 0, every y value on the graph of y = x2 is associated with two x-values.
Square Root Problems
One way to tackle basic square root problems by hand is to look for perfect squares "hidden" inside the problem. First, it's important to be aware of a few vital properties of squares and square roots. One of these is that, just as √x2 is simply equal to x (because the radical and the exponent cancel each other out):
That is, if you have a perfect square under a radical multiplying another number, you can "pull it out" and use it as a coefficient of what remains. For example, returning to the square root of 50
Sometimes you can wind up with a number involving square roots that is expressed as a fraction, but is still an irrational number because the denominator, the numerator or both contain a radical. In such instances, you may be asked to rationalize the denominator. For example, the number
has a radical in both the numerator and the denominator. But after scrutinizing "45," you may recognize it as the product of 9 and 5, which means that
Therefore, the fraction can be written
The radicals cancel each other out, and you are left with 6/3 = 2.