Next I will explain how to calculate the radicals or roots and I will also explain to you their definition, step by step and with examples and solved exercises.
To understand much better how to calculate the roots, as well as the rest of the course content in future lessons, you must first know how to deal with the powers perfectly.
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How to calculate radicals or roots
Before I begin to explain to you how to calculate radicals, you must know what the roots are and for that I will start by briefly reminding you what a power is.
A power, is indicated this way:
a = Power base
n = Power exponent
b = Power value
Which means that the 2 is multiplied 3 times and its result is 8.
So far you’ve got it all figured out, haven’t you?
For just as subtraction is the opposite operation to addition and division is the opposite operation to multiplication, so the root is the opposite operation to power.
Calculating a root consists of obtaining the base of the power, when you already know its exponent and its result.
Continuing with the previous example, we start from the value of the power and we want to calculate its base. So you have to ask yourself, what number elevated to 3 gives 8? Whose answer is 2.
Or put it another way, what is the cubic root of 8? The cubic root of 8 is equal to 2, because 2 raised to 3 (or to the cube) is 8:
Do you understand? The value of the potency is the content of the root (or radicating) and the exponent of the potency becomes the index of the root.
In general we have to:
Where for the general shape of the root:
n = Index
√ = Radical sign
b = Radicating
a = Root value
The sign of the root is called the radical. That’s why roots are also called radicals.
The square root is the opposite of squaring a number and has a 2 as its index, but on these roots, the index is not placed.
So when you see a root without an index, it’s already over-understood as a square root.
How to calculate the square roots?
For example, asking yourself what the square root of 81 is is the same as asking yourself: What square number does 81 give?
The square root of 81 is 9 because 9 squared is 81.
Square root solutions
A square root may have two solutions or no solution at all.
If the content of the root or radicating is positive, the root will have two solutions, one positive and one negative, but if the content of the root is negative, the root will have no real solution.
Not having a real solution means that it has no solution within the set of real numbers, since it does have a solution within the set of complex numbers, but since complex numbers are not the subject of this course, we will remain that the square roots of negative numbers have no real solution.
Let’s take a look at some examples of what I just explained:
The square root of 25 has two solutions: 5 and -5:
Because as much as 5 squared and -5 squared is 25:
However, the square root of -9 has no real solution:
There is no number that squared off as a result of a negative number
In general, any number (positive or negative) raised to an even exponent results in a positive number.
On the other hand, if the roots do not have a whole result, it is left in root form:
And only if strictly necessary, can you find its value with the calculator.
There is a method for calculating square roots by hand, but it is very laborious and impractical. The moment you stop using it you will forget it, so I recommend that you don’t try to learn it.
Cubic roots are roots that have index 3 and are the opposite operation to lifting a number to the cube.
We saw it in the first example:
The cubic root of 8 is 2 because 2 to the cube equals 8.
Cubic roots of negative numbers
Unlike square roots, cubic roots have a solution when the number is negative, because if there are negative numbers that when raised to the cube have a negative solution:
In general, a negative number high to an odd exponent has a negative result as well. For this reason, negative cubic roots do exist.
On the other hand, if the cubic root does not have a whole solution, it is left in the form of a root.
Now you practice. Proposed Exercises
The minus sign is multiplying the result of the root, which is positive. That is why the final result is negative.
The root solution is not whole, so it remains the same.
The minus sign is multiplying the result of the root, which is negative. That is why the final result is positive.
The root solution is not whole, so it remains the same.[/su_spoiler]
Now you have learned how to calculate the roots, both square and cubic, as well as those of any index.
Everything explained step by step, with exercises and examples solved.