The Basics of Solving a Lone Radical
A radical is a problem involving a sign that looks like a square root sign (see pictures). The square root is the most common radical for students to work with. It is a radical to the power of 2. Let’s talk about how to solve first a square root.
When you want to take the square root of a number, you have to ask yourself what number squared or what number multiplied by itself will equal the number I am trying to take the square root of. An example is the square root of 9. To find the square root of 9 we ask ourselves what number squared or what number multiplied by itself will equal 9. The answer is 3. 3^2 or 3 times 3 equals 9. Therefore, the square root of 9 is 3.
The same sort of logic is used to solve radicals of different powers. If you see a radical sign with a little number sitting right above the point or just to the left of it, then that number designates the power of that root. For example, if the number is 3, then instead of being a square root, this is a cube root. If the number is 5, then this is not a square root it is a fifth root. In order to solve a root to a different power then a square root, one must ask what number raise to that power equals the number that they are trying to take the root of. For example, if you are trying to find the cube root of 8, then you ask yourself, what number cubed or raised to the third power equals 8. The answer is that 2^3 = 8, so the cube root of 8 = 2. In another example, if you are looking for the fourth root of 256, then you ask yourself what number when raised to the fourth power equals 256? The answer is that 4^4 = 256, so the fourth root of 256 = 4.
You will see though this process that the opposite of taking a number to a certain power is taking the root of that power. For example 3^2, the square root of 3^2 (or 9) = 3 or in then other example, the fourth root of 4^4 (or 256) = 4
A lot of radical problems take some guess and check with the calculator. As a hint, start small! Most of the time, the answer will be 2 or 3, especially when you are first starting this subject.
Multiplying Radicals
When you are multiplying two radicals of the same power, then you can combine the numbers under one big radical. For example, (the square root of 27) times (the square root of 3) equals the square root of (27 times 3), which becomes the square root of 81, which equals 9.
This operation can only be performed when multiplying roots of the same power. If you are multiplying a cube root times a fifth root, then they cannot be combined this easily.
Dividing Radicals
If you are dividing radicals of the same power, then you can combing the division under one radical. For example if you are trying to solve (the cube root of 135)/(the cube root of 5), then you can change it to be the cube root of (135/5), which becomes the cube root of 27, which equals 3.
Just like with multiplication, this can only be done if both radicals are to the same power.
Simplifying Radicals with Non-Perfect Roots
Sometimes you will be given a root that doesn’t have a very pretty answer. For example the square root of 27. There is no perfect square or number without a decimal that solves this answer. To simplify this, without using decimals, we check to see if 27 is divisible by perfect square or in other words if one or more of the factors of 27 is a perfect square. In this case 9 can be divided out of 27 and is a perfect square. You would first rewrite 27 as 9 times 3. Just like how we talked about being able to combine two radicals that are multiplied to be two numbers multiplied under a radical, we can also go the opposite direction. You will next change the square root of (9 times 3) to be (the square root of 9) times (the square root of 3), which becomes 3 times (the square root of 3), which is where you will leave the answer for that problem.
You can perform this operation multiple times if at first you do not find the largest perfect root to pull out. For example, if you were given the fifth root of 15552. You would first notice that 15552 is an even number, thus it is highly likely that it will be divisible by 2^5. 2^5 =32 and we can divide 15552 by 32 so we see that we can change the fifth root of 15552 to be the fifth root of (32 times 486), which then can become (the fifth root of 32) times (the fifth root of 486). We know that since 2^5 = 32 that the fifth root of 32 = 2, so we now have 2(the fifth root of 486), 486 is also an even number, but this time we see that 32 is not a factor of 486. So, we next check 3^5 = 243, and we find that 243 is a factor of 486. So we can change 2(the fifth root of 486) to be 2[the fifth root of (243 times 2)], which then becomes 2(the fifth root of 243) times (the fifth root of 2), which becomes 2(3)times (the fifth root of 2), and lastly we have 6(the fifth root of 2).
Adding and Subtracting Radicals
When adding and subtracting radicals, the most important first step is to make sure that all of the radicals you are trying to add and subtract are as simplified as possible. Then, you can only add and subtract like terms. For example, you can only add amounts of square roots of 3 with other square roots of 3. For this not only do the powers of the radicals need to match, but also the value of what you are taking the root of in order to be able to combine terms.
Here is an example problem:
(square root of 50) + (square root of 2) – (cube root of 54) – 2(cube root of 2), using the rules mentioned above for simplifying radicals we find that we have: 5(square root of 2) + (square root of 2) – 3(cube root of 2) -2(cube root of 2). We can combine the square roots of 2 the same way you would combine something like 5x + x. We say to ourselves, I had 5 square roots of 2 and not I am adding one more so I know have 6(square root of 2). The cube roots of 2 can be combined in a similar way. I’m missing 3 cube roots of 2, then I lose 2 more cube roots of 2, so how many total negative cube roots of 2 do I have? -5(cube root of 2), so our final answer is 6(square root of 2) – 5(cube root of 2). Even though in both cases we are taking a root of 2, these two answers cannot be combined, because the roots are of different powers.
