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To use this online tool, you simply need to specify the base of the power and its exponent, whether it's a positive integer, a fractional one, or even a negative one.
Once you have entered all the requested fields, click on “Calculate”, and the calculator will return the result of the exponentiation within seconds.
Powers are simply repeated multiplications and are formed by two elements: the exponent (n) and the base (a). The latter is the number that we will raise to the former.
$$a^n$$
When dealing with integers for simplicity, imagine that you have to multiply a number by itself several times, for example, multiplying the number 3, 4 times, it would be too laborious to write the operation like this:
$$3 \times 3 \times 3 \times 3$$
To speed up this mathematical process, you could use the power, which is nothing but a reformulation of the previous operation, redone in a simpler, faster, and more convenient way:
$$3 \times 3 \times 3 \times 3 = 3^4$$
If you want to express 3^{3} verbally, the most appropriate way is to say “three raised to three” or “three raised to the third power” or the most used “three cubed”. If it were raised to 2, you should say “three squared”.
In summary: when we operate with integers, we know the power of a number as the multiplication of this number n times. Such operation is indicated with a^{n}, where a is the base and b is the exponent.
When you have to deal with an operation involving powers, for example: 5^{3}, the first thing you should do is identify the two parts that make it up.
The “bottom” number, in our example 5, is the base, while the “top” number is the exponent.
If the exponent is 0 and the base is an integer, the result will always be 1. For example:
$$5^0 = 1, \quad 18522^0 = 1$$
Powers whose exponent is 1, will remain exactly as the base.
To execute the calculation of a power, simply multiply the base by itself the number of times the exponent indicates, continuing with our example 5^{3}:
$$5 \times 5 \times 5 = 125$$
If you want to perform the operation by hand, or even in your head, our recommendation is to do it by multiplying the numbers two by two, that is:
$$5 \times 5 = 25; \quad 25 \times 5 = 125$$
If you want to add or subtract powers, you will need as a fundamental requirement that both have the same base. This operation is a simplification.
We could affirm that:
$$6^3 + 6^3$$
Is equal to:
$$ (1)(6^3) + (1)(6^3)$$
Therefore, the result of the sum of these two powers would be equal to:
$$2(6^3)$$
If we add the results of the powers individually, it would be the same but twice, therefore, by multiplying it by two we would be simplifying the process.
It may seem a bit cumbersome and hard to digest, but with an example, it will surely become much clearer:
$$6^3 + 6^3 = (6)(6)(6) + (6)(6)(6) = 2 (6)(6)(6) = 2 6^3$$
For subtractions, the procedure would be practically the same as in addition, only that the numbers preceding the powers would be subtracted instead of added.
To multiply powers, it is essential that all of them have, at least, the base in common, then you simply add the exponents of all of them.
It is actually another simplification, let's go with an example:
$$(5^2)(5^4)(5^3)$$
Which is the same as:
$$[5\times5] [5\times5\times5\times5][5\times5\times5] = 5^9$$
If you need to multiply different exponents within parentheses, as it could be
$$ (4^2)^3$$
simply multiply both exponents among them, and you will obtain the definitive exponent, for example:
$$(4^2)^3 = (4^2)(4^2)(4^2) = 4^6 $$
As the base is always the same, you can add the different exponents to calculate the result.
When dealing with a power whose exponent is negative, you must convert the exponent into a positive number, and for that, you will need to reformulate it in the form of a fraction.
Guide yourself through the following examples:
$$4^{(-2)} = \frac{1}{4^2}$$
$$3(4^{(-2)}) =\frac{3}{4^2}$$
To divide two powers with the same base subtract the exponents from each other. As division is the operation just opposite to multiplication, it is solved by doing just the opposite procedure.
If we encounter a power whose exponent is written fractionally, as, for example, 4^{1/2} we just need to know that it is as if it were the square root of that number, as long as it is raised to 1, that is:
$$4^{\frac{1}{2}}=\sqrt{4}=2$$
This premise can be extrapolated to all the fractional exponents you encounter.
If for example you find the number 8 raised to 1/3, it means that to solve it you will need to calculate the cube root of 8, in this case, the result would be 2.
This is fundamentally because raising to the power is the opposite procedure to the root.
When you are working with powers, one of the things you should keep in mind is that increasing the exponent of any of them will greatly increase the final result of the operation.
Therefore, even if the final result seems to be a disproportionate figure, do not reject it for that reason, as it could be correct.
On the other hand, do not forget that a power raised to 1 means that the whole number remains as it is, and if, on the other hand, it is raised to 0 it will always be 1.
To conclude, you should remember that most calculators have at least one key to solve powers.
This function can be found in two ways, either with the command exp or using the key x^{n}.
If your physical calculator is very old and does not have this function, you can always use one of ours which are also free.
And that's all you need to know to perform operations with powers, we have made the explanation in the simplest way, always step by step and guided with examples so that no one gets lost along the way.
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