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In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. If youre seeing this message, it means were having trouble loading external resources on our website. If youre behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Operations with cube roots, fourth roots, and other higher-index roots work similarly to square roots, though, in some spots, well need to extend our thinking a bit. Ill explain as we go. Simplifying Higher-Index Terms In the previous pages, we simplified square roots by taking out of the radical any factor which occurred in sets of two. For the second root, we needed a second copy. For higher-index roots, the thinking is the same. If we have a cube root, we can take out any factor that occurs in sets of three; in a fourth root, we take out any factor that occurs in sets of four; in a fifth root, we take out any factor that occurs in sets of five; and so forth. For instance: Square Roots and Cube Roots Previously, I could pull from a square root anything that of which I had two copies. In the same way now, I can pull from a fourth root anything of which Ive got four copies. Since 16 = 24, then: Im taking the cube root. I can then pull out of the radical any factor that occurs three times. Since 8 = 23, then this radical will simplify completely. My first step is to factor this completely: 54 = 2 · 27 = 2 · (3 · 3 · 3) I have three copies of 3, so I can pull a 3 out of the cube root, leaving a 2 inside. Again, I start with factoring: 48 = 3 · 16 = 3 · 2 · 2 · 2 · 2 I have four copies of the factor 2, but this is a cube root, so I can only pull out a 2 for three of these copies. The 3 and the fourth 2 will remain behind inside the radical. I know that 27 = 33, so the cube root will simplify to a whole number. Then Ill finish by multiplying. Theyve given me variables inside this radical, but the process works the same way as always. Im taking a fifth root, so I can pull out of the radical anything for which Ive got five copies. The 32 is 25, so this will come out of the radical. The x10 = (x2)5, so an x2 will come out. The y6 = (y5)(y1), so Ill be able to pull a y out, while leaving the last y inside the radical. And z7 = (z5)(z2), so Ill be able to pull a z out, while leaving a z2 inside. My work looks like this: Note: When youre simplifying radical expressions with variables, if the radical is an even-index root (like a square root or a fourth root), theyll probably specify that you should assume that all variables are non-negative (or positive). This is to avoid having to take into consideration whether or note absolute-value bars are necessary on your answer. If you arent sure what Im talking about, check here. Multiplying Higher-Index Roots This multiplication works just like the multiplication of square roots, in that the product of two of the same higher-index root can be converted to the higher-index root of the product. Then I simplify in the usual manner. In this case, theyve given me a product of fourth roots. I can convert the product of radicals into the radical of a product. Then I can simplify. Adding Higher-Index Roots The terms in this expression are both cube roots, but I can combine them only if theyre the cube roots of the same value. Right now, they arent. So Ill simplify the radicals first, and then see if I can go any further. I note that 8 = 23 and 64 = 43, so I will actually be able to simplify the radicals completely. I cant simplify the second radical at all. But I can simplify the first radical, because 81 = 34 = (33)(3). So I will end up with the sum of two third roots of three, which I can combine. Dividing Higher-Index Roots The denominator is a cube, being 27 = 33, so I can easily simplify and arrive at a rationalized denominator: This is similar to the previous exercise, but here the cube (that is, the 27) is in the numerator. I cant simplify this expression properly, because I cant simplify the radical in the denominator down to whole numbers: To rationalize a denominator containing a square root, I needed two copies of whatever factors were inside the radical. For a cube root, Ill need three copies. So thats what Ill multiply onto this fraction. I have one copy of the factor 5 in the denominator. Ill multiply, top and bottom, by the cube root of 25, which will provide the additional two copies of 5 that I need, in order to rationalize the denominator. This final expression is arguably not much simpler than the original expression. In this context, simplify meant rationlize the denominator. Oftentimes, the right answer wont be much, if at all, simpler than what youd started with. Since 72 = 8 × 9 = (2 × 2 × 2) × (3 × 3), I have only three 2s and two 3s. In other words, as the fraction currently stands, I wont have enough of any of the denominators factors to get rid of the radical. To take anything out of a fourth root, I would need four copies of each factor. For this denominators radical, Ill need two more 3s and one more 2. So Ill multiply, top and bottom, by the fourth root of 3 · 3 · 2 = 18. You can use the Mathway widget below to practice working with higher-index expressions. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathways. Please accept preferences cookies in order to enable this widget. (Click Tap to view steps to be taken directly to the Mathway site for a paid upgrade.) URL: https: Page 1Page 2Page 3Page 4Page 5Page 7
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