All cube roots asked on this sheet are exact and will always be less than 100. [br]We can use this as a clue to mental solve them really fast.[br]If you do not know any [b]trick[/b], [b]down below[/b] we have the explanation of a very simple one.

[*]Remember it is a matter of practicing mental calculation, so try not need to write anything to help you working this out.[/*][list][*]You can use the mental strategies explained below, or any other one you know.[/*][*]But it is really important for you to [b]know why[/b] the trick works. These exercises are intended to help us memorizing the [b]first perfect cubes[/b] and improve our mental skills, while we are using some other mathematical knowledge.[br][br][/*][/list][list][*]In order to gain time, consider that it is not neccesary for you to correct each exercise one by one. You can wait until finishing, when all the activities will be corrected at the same time.[/*][*]Before starting, we can choose the available time to fill the worksheet.[br][br][/*][*]Each sheet will be asigned a mark out of 100 points after finishing it.[br][list][*]first 5 correct calculations are worth 10 points each.[/*][*]from that on, each will be worth 5 points.[/*][*]also, each mistake cancels one correct answer.[/*][*]if you try several sheets, your highest mark will be kept.[br][/*][/list][br][/*][/list]

When the radicand is less than 1000 000, its cube root is less than [math]\sqrt[3]{1000\,000}=100[/math].[br][br]Therefore, the [b]cube root[/b] will only have [b]two digits[/b], and can be written as [math]10a+b[/math], and the radicand must be something like [math](10a+b)^3=a^3\cdot 1000+\ldots+b^3[/math].[br]Let us take a look at the cube of the first nine numbers. ¡Curious, the last digits are all different![br][center]1[sup]3[/sup]=[b]1[/b], 2[sup]3[/sup]=[b]8[/b], 3[sup]3[/sup]=2[b]7[/b], 4[sup]3[/sup]=6[b]4[/b], 5[sup]3[/sup]=12[b]5[br][/b]6[sup]3[/sup]=21[b]6[/b]=3, 7[sup]3[/sup]=34[b]3[/b], 8[sup]3[/sup]=51[b]2[/b], 9[sup]3[/sup]=72[b]9[/b].[/center]Keeping this information in mind, we can analizar how to obtain the root:[br][list][*]The [b]tens digits[/b] must be the cube root of the first three left digits of the radicand (two if there are only 5 digits, and one if there are only 4 in the radicand), since it is the part multiplied by 1000. [br]Look out! as there are also some other terms in that sum, (indicated with ellipsis), they can add some number to the thousands, so it is not neccesary the exact root.[/*][*]Term [math]b^3[/math] gives us the [b]units digit[/b]. Provided that only one cube ends in that digit, we only need to check which one is it to obtain the units digit of the root.[/*][/list]

Find the cube root of:[br][list][*][b]3375[/b][br]Splitting the digits, we get 3 275, so[list][*]the tens digit of the root is the cube root of 3, which is 1.[br][/*][*]As the radicand ends in 5, the units digit must be 5, since it is the only one whose cube ends in 5.[/*][*]So, the solution is [math]\sqrt[3]{3375}=15[/math].[/*][/list][/*][*][b]12167[/b][br]Splitting the digits, we get 12 167, so[list][*]the tens digit of the root is the cube root of 12, which is 2, since 2[sup]3[/sup]=8.[br][/*][*]As the radicand ends in 7, the units digit must be 3, since it is the only one whose cube ends in 7.[/*][*]So, the solution is [math]\sqrt[3]{12167}=23[/math].[br][/*][/list][/*][*][b]226981[/b][br]Splitting the digits, we get 226 981, so[list][*]the tens digit of the root is the cube root of 226, which is 6, since 6[sup]3[/sup]=216.[br][/*][*]As the radicand ends in 1, the units digit must be 1.[/*][*]So, the solution is [math]\sqrt[3]{226981}=61[/math].[/*][/list][/*][/list]

[size=85][list][*]Images by [url=https://programacrea.educarex.es/]Programa CREA[/url] from Extremadura (Spain).[/*][*][url=https://www.geogebra.org/m/t8nzndzd]Original resource in Spanish (link)[/url].[br][/*][/list][/size]