Roots

Learning Objectives

• Find principal square roots and their opposites.
• Approximate square roots and find exact roots with a calculator.

You have probably dealt with the roots of plants and trees when gardening, but did you know that there are roots in mathematics, too?

Yes, roots do exist in math. The most common root is the square root A number that when multiplied by itself gives the original nonnegative number. For example, `6 * 6 = 36` and `-6 * -6 = 36` so `6` is the positive square of `36` and `-6` is the negative square root of `36`. . Let’s take a look at what roots are, how they relate to exponents, and how you find the square root of a number.

To help understand square roots, let’s review some facts about exponents. Look at the table below.

You can think of exponential numbers as “repeated multiplication.”

Just as dividing is the inverse of multiplying, the inverse of raising a number to a power is taking the root of a number. The most common root (and the one you will be studying here) is called the square root. When you are trying to find the square root of a number (say, `25`), you are trying to find a number that can be multiplied by itself to create that original number. In the case of `25`, you can find that `5 * 5 = 25`, so `5` must be the square root.

The symbol for the square root is called a radical symbol The symbol, `sqrt" "`, used to denote the process of taking a root of a quantity. and looks like this: `sqrt" "`. The expression `sqrt25` is read “the square root of twenty-five” or “radical twenty-five.” The number that is written under the radical symbol is called the radicandThe number or value under the radical symbol.. Take a look at the following table.

Look at `sqrt25` again. You may realize that there is another value that, when multiplied by itself, also results in `25`. That number is `-5`.

`5*5=25`

`-5*-5=25`

By definition, the square root symbol always means to find the positive root, called the principal root The positive square root of a number, as in `sqrt(16)=4`. By definition, the radical symbol always means to find the principal root. Note that zero has only one square root, itself (since `0 * 0 = 0`). . So while `5 * 5` and `-5 * -5` both equal `25`, only `5` is the principal root. You should also know that zero is special because it has only one square root: itself (since `0 * 0 = 0`).

If you know the principal square root, you can also find its opposite. (Recall that any number plus its opposite will equal `0`. So, for instance, `a+(-a)=0`.) In the table below, notice that while `sqrtx` will give the principal root, `sqrt(-x)` will give its opposite. For example, `sqrt36` equals the principal square root, `6`, and `-sqrt36` equals its opposite, `-6`.

There you have it: putting a negative sign in front of a radical has the effect of turning the principal root into its opposite (the negative square root of the radicand).

So now you have reviewed exponents, and you have been introduced to square roots. How does knowing about one help you understand the other?

Exponents and roots are connected because roots can be expressed as fractional exponents. For now, let’s look at the connection between the exponent “`1/2`” and square roots; you will learn about other fractional exponents and other roots later on. The square root of a number can be displayed by using the radical symbol or by raising the number to the `1/2` power. This is illustrated in the table below.

The square of `4` is `16` because `4` times `4` is equal to `16`. Recall from your work with exponents that this can also be written as `4^2=16`.

Thinking this way, you can identify that the square root of `9` is `3` because `3 * 3 = 9`. Similarly, the square root of `25` is `5` because `5 * 5 = 25`, and the square root of `x^2` is `x` since `x*x=x^2`. For example, `sqrt(7^2)=7`. (You will often see this type of notation, where you take the square root of a squared number, when you simplify, multiply, and divide radicals.)

Square roots and exponents are connected. Keep that in mind as you begin simplifying some square roots.

This first example, “Simplify `sqrt144`,” can be read “Simplify the square root of `144`.” Think about a number that, when multiplied by itself, has a product of `144`.

However, if the negative sign is under the radical as in `sqrt(-49)`, there is no way to simplify it using real numbers. That is because there is no number that you could multiply by itself to get `-49`. Remember, a negative number multiplied by a negative number results in a positive number: `-7 * -7 = 49`.

If finding the square root by trial and error is difficult, you can use what you know about factoring to help you determine the principal root.

Notice something that happened in the final step of this example: the expression `sqrt(2*2*2*2*3*3)` was rewritten as `sqrt(12*12)`, and then `sqrt(12^2)`. You split the factors into identical groupings, multiplied them, and arrived at a squared number.

Many times, though, it is easier to identify factor pairs after you have gone through the process of factoring the original radical. For example, look at `sqrt(2*2*2*2*3*3)` again. How many pairs of `(2*2)` do you see? What about `(3*3)`? If you could somehow identify smaller squared numbers underneath the radical instead of recombining all the factors (as you did when you found that `sqrt(2*2*2*2*3*3)=sqrt(12*12)`), you might be able to simplify radicals more quickly.

This is where it helps to think of roots as fractional exponents. Recall the Product Raised to a Power Rule The product of two or more non-zero numbers raised to a power equals the product of each number raised to the same power: `(ab)^x = a^x * b^x` from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written `(ab)^x=a^x*b^x`. So, for example, you can use the rule to rewrite `(3x)^2` as `3^2*x^2=9*x^2=9x^2`.

Now instead of using the exponent `2`, let’s use the exponent `1/2`. The exponent is distributed in the same way.

`(3x)^(1/2)=(3)^(1/2)*(x)^(1/2)`

And since you know that raising a number to the `1/2` power is the same as taking the square root of that number, you can also write it this way.

`sqrt(3x)=sqrt3*sqrtx`

You can think of any number underneath a radical as the product of separate factors, each underneath its own radical. Using this idea helps you identify smaller squared numbers, which often lets you simplify radicals more quickly.

This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with `sqrt((2*2)(2*2)(3*3))`, you can rewrite the expression as the product of multiple perfect squares: `sqrt(2^2)*sqrt(2^2)*sqrt(3^2)`. Let’s take another look at `sqrt144` using this new idea.

You get the same solution in both cases, but it is often easier to pull out smaller factor pairs and then multiply them together (as shown here) than recombining all the factors to find the full root (as shown in the first example).

So far, you have seen examples that are perfect squares. That is, each is a number whose square root is an integer. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a power.

The final answer `3sqrt7` may look a bit odd, but it is in simplified form. You can read this as “three radical seven” or “three times the square root of seven.”

Another approach to handling square roots that are not perfect squares is to approximate them by comparing the values to perfect squares. Suppose you wanted to know the square root of `17`. Let’s look at how you might approximate it.

This approximation is pretty close. If you kept using this trial and error strategy, you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand.

For this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key (`sqrt" "`) that will give you the square root approximation quickly. On a simple `4`-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key.

Try to find `sqrt17` using your calculator. Note that you will not be able to get an “exact” answer because `sqrt17` is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, `sqrt17` is approximated as `4.123105626`. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren’t perfect squares.

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of `0` is `0`. You can only take the square root of values that are greater than or equal to `0`. The square root of a perfect square will be an integer. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root. Square roots can be approximated using trial and error or a calculator.