Simple and not so ways how to calculate the cube root
How many angry words uttered in his address? Sometimes it seems that the cube root is incredibly different from the square one. In fact, the difference is not so great. Especially if you understand that they are only special cases of the common root of the n-th degree.
But with its extraction problems may arise. But most often they are associated with cumbersome calculations.
What you need to know about the root of an arbitrary degree?
First, the definition of this concept. The n-th root of some “a” is the number that, when raised to the power of n, gives the original “a”.
And there are even and odd degrees at the roots. If n is even, then the radical can be only zero or a positive number. Otherwise, there will be no real answer.
When the degree is odd, then there is a solution for any value of "a". It may well be negative.
Secondly, the root function can always be written as a degree, which is a fraction. Sometimes it is very convenient.
For example, “a” to the 1 / n degree will be the nth degree root of “a”. In this case, the base of the degree is always greater than zero.
Similarly, “a” to the degree n / m will be represented as the root of the mth degree from “an».
Thirdly, all actions with degrees are valid for them.
- They can be multiplied. Then the exponents add up.
- The roots can be divided. The degree will need to be deducted.
- And raise to a power. Then they should be multiplied. That is, the degree that was, to the one to which they are building.
What are the similarities and differences of square and cubic roots?
They are similar, like brothers, only the degree they have is different. And the principle of calculating them is the same; the only difference is how many times the number must multiply by itself in order to get the radical expression.
And about the significant difference was mentioned a little higher. But repeat will not be superfluous. The square is extracted only from a non-negative number. While it is easy to calculate the cube root from a negative value.
Cubic root extraction on a calculator
Every person at least once did this for a square root.And what if the degree of "3"?
On a regular calculator, there is only a button for a square, but a cubic one is not. This will help a simple enumeration of numbers that are multiplied by three times. Did you get a radical expression? So this is the answer. Did not work out? Pick up again.
And what about the engineering form of a calculator in a computer? Hooray, there is a cubic root. This button can be simply clicked, and the program will give an answer. But that's not all. Here you can calculate the root of not only 2 and 3 degrees, but also any arbitrary. Because there is a button in which the degree of the root is "y". That is, after pressing this key, you will need to enter another number, which will be equal to the degree of the root, and only then “=”.
Manual cube root extraction
This method will be required when the calculator is not at hand or you cannot use it. Then in order to calculate the cube root of a number, it will take an effort.
First, see if the full cube is not obtained from any integer value. Maybe the root is 2, 3, 5 or 10 in the third degree?
Otherwise, you will need to be considered a column. The algorithm is not the easiest. But if you practice a little, the actions will be easily remembered.And calculating the cube root will no longer be a problem.
- Mentally divide the radical expression into groups of three digits from the decimal point. Most often the fractional part is needed. If it is not there, then you need to add zeros.
- Determine the number whose cube is less than the integer part of the radical expression. Write it in the intermediate answer above the root sign. And under this group to place its cube.
- Perform subtraction.
- Add the first group of digits to the remainder.
- In the draft, write the expression: a2* 300 * x + a * 30 * x2+ x3. Here “a” is an intermediate answer, “x” is a number that is less than the resulting balance with numbers assigned to it.
- The number "x" must be written after the comma of the intermediate answer. And the value of the entire expression to write under the compared balance.
- If accuracy is sufficient, then the calculations stop. Otherwise, go back to item number 3.
An illustrative example of the calculation of the cubic root
It is needed because the description may seem complicated. The figure below shows how to extract the cube root of 15 to the nearest hundredth.
The only difficulty this method has is that with each step the numbers increase many times and it becomes more difficult to count in the bar.
- 15> 23, then under the integer part 8 is written, and 2 is above the root.
- After subtracting from 15 eight, we get the remainder 7. Three zeros must be assigned to it.
- a = 2. Therefore: 22* 300 * x +2 * 30 * x2+ x3<7000, or 1200 x + 60 x2+ x3< 7000.
- The selection method turns out that x = 4. 1200 * 4 + 60 * 16 + 64 = 5824.
- Subtraction gives 1176, and the number 4 appears above the root.
- Assign three zeros to the remainder.
- a = 24. Then 172800 x + 720 x2+ x3< 1176000.
- x = 6. The calculation of the expression gives the result 1062936. Balance: 113064, above the root 6.
- Assign zeros again.
- a = 246. Inequality is obtained as follows: 18154800х + 7380х2+ x3< 113064000.
- x = 6. Calculations give the number: 109194696, Balance: 3869304. Above the root 6.
The answer is the number: 2, 466. Since the answer must be given to the hundredths, it needs to be rounded off: 2.47.
Unusual way to extract cubic root
It can be used when the answer is an integer. Then the cubic root is extracted by decomposing the radicand into odd terms. Moreover, such terms should be the minimum possible number.
For example, 8 is represented by the sum of 3 and 5. A 64 = 13 + 15 + 17 + 19.
The answer will be a number that is equal to the number of terms. So the cubic root of 8 will be equal to two, and from 64 - four.
If the root value is 1000, then its decomposition into terms will be 91 + 109 + 93 + 107 + 95 + 105 + 97 + 103 + 99 + 101. A total of 10 terms. This is the answer.