Examples of how to solve the power of a fraction product

Perhaps the most convenient thing to do, before approaching each of the cases that can serve as examples to the correct way in which any operation that raises to an exponent the operation of sustained product between two fractions must be resolved, is to briefly revise the very definition of this operation, in order to understand each one of these examples in its precise context.

Power of a product of fractions

However, perhaps it is best to remember first that Mathematics defines fractions as all mathematical expressions, used to represent non-integer or non-exact quantities. Likewise, this discipline warns that fractions will always be composed of two elements: numerator, which will occupy the upper part of the expression, indicating how many parts of the whole have been taken; and denominator, which will serve to indicate into how many parts this whole is divided.

As for the operation of power of a product of fractions, basically it can be said that it is a procedure consisting in elevating to a certain exponent a multiplication operation, where the factors are powers. In this sense, this product operation must be multiplied by itself, as many times as the corresponding exponent indicates, in order to comply with the condition of abbreviated multiplication of the potentiation operation.

Steps to solve a fraction product power

Likewise, Mathematics indicates that this type of procedure must be solved according to a specific method, consisting of a group of steps, which will be completed in the following order:

  1. In the first place, it is necessary to begin by knowing the elements and characteristics that compose each one of the fractions that comprise the operation of the product.
  2. Likewise, the value of the exponent to which this operation is elevated will be taken into account.
  3. Each fraction will then be elevated to the exponent to which the product operation has been elevated.
  4. Once this has been done, each rational-based potentiation operation obtained will be resolved separately, which will be done by elevating each element to the pertinent exponent.
  5. Once obtained the fractions, product of the respective powers of rational base, it will be necessary to proceed to multiply both fractions, remembering that the numerator of one must be multiplied by the denominator of the second one, while it will be made equal with the denominators of each fraction.
  6. Finally, we will try to simplify the fraction obtained.

Likewise, the mathematical discipline indicates that the correct way to solve this operation can be expressed in the following way:

Examples of how to solve the power of a product of fractions

However, perhaps the most effective way to study this type of operations is through the exposition of some examples that allow us to see in a practical way how each of the steps indicated by Mathematics are fulfilled. Here are some exercises on how to find the power of the product of fractions:

Example 1

Resolve the next operation:

Once this approach has been made, each fraction will then be elevated to the exponent to which the operation of the product they both constitute has been elevated:

In the second place, each one of the powers of rational base that have been generated will then be resolved:

Once this result is obtained, the multiplication of fractions must be continued:

Once the product operation is resolved, the fraction will be simplified:

When the simplest possible form is reached, the operation is assumed to be resolved.

Example 2

Resolve the next operation:

In this case you have a product operation elevated to a negative exponent, to solve this operation you must elevate each multiplication factor to this exponent:

At this point, prior to elevating each fraction to its respective exponent, it is best to make the exponent go from negative to positive, which will be achieved by inverting the elements of each operation:

Having now a positive exponent, each one of the rational-based power operations will be solved:

At this point the multiplication of fractions must be continued:

The operation will then be simplified:

What could have been done by suppressing the 27 found in both factors. Without being able to simplify the operation further, the operation will be considered resolved.

Example 3

Resolve the next operation:

When analyzing each one of the elements that constitute this operation, we will have to deal with a multiplication of fractions elevated to zero. We will begin, according to the procedure indicated by Mathematics, to elevate each of the multiplication factors to this exponent:

When doing so, a solution must then be given to each rational base power; however, since each one of them is elevated to zero, one must remember the mathematical law that states that whenever a fraction is elevated to the exponent zero it will result in unity:

In the same way, seeing in principle that the operation of the product was elevated to this exponent, this result could be interpreted without the need to elevate each factor of the multiplication to this number, in this way it could have simply been solved in this way:

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Examples of how to solve the power of a fraction product
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phoneia.com (October 25, 2019). Examples of how to solve the power of a fraction product. Bogotá: E-Cultura Group. Recovered from https://phoneia.com/en/education/examples-of-how-to-solve-the-power-of-a-fraction-product/