In my history book the expression square root (with a 3 in the nook( around a square root of ten is given. It says it equals the square root of ten with 6 in the nook. However, I don't understand where the six came from. I thought you multiply the two things in the nooks together. So 3 and 1 (since nothing was written in the nook of ten) would equal three. I thought it would be the radical of ten with a three in the nook. How did I go wrong?
square root has a understood 2 in the nook, it is to the square root. 3 x 2 is six.
It seems like you are referring to the notation used for expressing the cube root of a number. While it is true that multiplying the number in the "nook" (the little number written to the left of the radical symbol) by the number inside the radical gives you the answer, in this case, the cube root is slightly different from the regular square root.
The cube root, denoted by the symbol ∛, finds the number that, when multiplied by itself three times, gives the value inside the radical. For example, the cube root of 8 (∛8) is 2 because 2 × 2 × 2 equals 8.
In your case, if you have the expression ∛10, the cube root of 10, there is no number written inside the nook. In situations like this, the default assumption is that the number inside the nook is 3. So, ∛10 means the cube root of 10 with a 3 in the nook.
To find the cube root of 10, you would need to find a number which, when multiplied by itself three times, gives you 10. Since there is no whole number that satisfies this condition, we usually express the cube root of 10 as an irrational number, approximately 2.154. Therefore, stating ∛10 equals the square root of 10 with 6 in the nook is not accurate.
Remember, when you see ∛10, it means the cube root of 10, and the number inside the nook indicates that the radical operation is being applied three times.