Product of rational powers of equal base

Perhaps best of all, prior to addressing an explanation of the correct way to resolve an operation that raises the product of fractional powers that match its bases, is to briefly review some definitions, which will allow us to understand this procedure in its just mathematical context.

Fundamental definitions

In this sense, it may also be prudent to focus this theoretical review on three specific concepts: Power-up, Fractions and Rational-Based Powers, as it is the notions directly related to the operation that involves multiplying powers of fractions, where different factors have equal basis. Here are each of these definitions:

Empowerment

In this way, it will begin by saying that the different mathematical sources generally choose to mark the Empoweror as an abbreviated multiplication, which will then aim to determine what is the product that is obtained by multiplying a number by itself as many times as the second numeric element involved in the operation points out. This operation can be represented mathematically as follows:

aⁿ = aⁿ₁ . aⁿ₂. aⁿ₃…

Likewise, mathematical discipline states that the Empoweror may be understood as an operation consisting of three elements, explained in turn as shown below:

Base: First, the Base will be understood as the numerical element intended to multiply itself, every time it dictates the second number that participates in the operation.
Exponent: For its part, the Exponent will be the number that tells the base how many times it should multiply on its own.
Power: Finally, the Power is interpreted as the final result of the operation, that is, the product that is obtained by multiplying the base every time indicated by the exponent.

Fractions

It will also be desirable to take a moment to reflect on the notion of Fractions, which are understood as those mathematical expressions, through which you realize rational or fractional numbers, that is, that fractions are used always to represent non-exact or non-whole amounts. Likewise, the mathematical discipline has pointed out that fractions are made up of two elements, explained as follows:

Numerator: on the one hand, the Numerator will constitute the top of the fraction. Your mission will be to point out how many parts of the whole have been taken or represented by this expression.
Denominator: Second, the Denominator will be the element that occupies the bottom of the fraction, having the task of indicating how many parts the whole is divided into.

Rational-based powers

Finally, it will also be necessary to approach the definition of Rational-based Powers, which have been generally explained by mathematics as a type of empowering operation in which the base consists of a rational number or fraction. As in whole-base fractions, the solution of this operation will be achieved by multiplying the base by itself the times the exponent points to:

However, most sources are inclined to solve these operations through the general formula for fractional powers, which indicates that it is best in these cases to raise each element of the fraction separately to the indicated exponent, procedure which may be represented mathematically as seen below:

Fraction power product of the same base

Once these definitions have been revised, it is perhaps certainly much easier to address an explanation of how any multiplication of rational base powers where it is determined that they all have the same basis, i.e., with the same fraction as the basis.

In such cases, there is a mathematical law that indicates – that as with the entire base powers – one should choose to assume a single base and proceed to add each of the exponents presented. This property may be expressed mathematically as follows:

Example

However, perhaps the best way to complete an explanation of how such an operation should be resolved is to expose a particular example, which allows us to see in practice how the mathematical property that dictates how to proceed in the if the factors of a fractional power multiplication have equal bases. Here’s the following exercise:

Solve the following multiplication of fractional powers:

In order to comply with the postulate, it must be verified that in fact both powers have the same base, so then we will proceed to assume only one of them, and add their exponents:

Having done this, it must then continue to solve the operation, for which the general formula of fractional powers will be applied, raising each element of the expression to the indicated exponent:

At the power result, if it is determined that it cannot be further simplified, then this product will be assumed as the final solution of the operation.

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