Question

Perform the indicated operation of multiplication or division on the rational expression and simplify

Answer 1

Operation on rational expressions

Step 1: division of fractions

The division of two fractions is the same as multiplying the first by the inverted second fraction:

Then, in this case:

`(24y^2)/(5x^2)/(6y^3)/(25x^2)=(24y^2)/(5x^2)*(25x^2)/(6y^3)`

Step 2: multiplication of two fractions

We multiply two fractions by multiplying the numerators and the denominators:

`(24y^2)/(5x^2)*(25x^2)/(6y^3)=(24y^2*25x^2)/(5x^2*6y^3)`

Step 3: simplifying the numbers of the fraction

We know that

`(25)/(5)=5\text{ and }(24)/(6)=4`

Then, we can use this in our fraction:

`\begin{gathered} (24y^2*25x^2)/(5x^2*6y^3)=5\cdot4(y^2x^2)/(x^2y^3) \\ \downarrow\text{ since 5}\cdot4=20 \\ 5\cdot4(y^2x^2)/(x^2y^3)=20(y^2x^2)/(x^2y^3) \end{gathered}`

Step 4: exponents of the result

We know that if we have a division of same base expressions (same letters), the exponent is just a substraction:

`\begin{gathered} (y^2)/(y^3)=y^(2-3)=y^(-1) \\ (x^2)/(x^2)=x^(2-2)=x^0=1 \end{gathered}`

Then,

`20(y^2x^2)/(x^2y^3)=20y^(-1)\cdot1=20y^(-1)`

Since negative exponents correspond to a division, then we can express the answer in two different ways:

`20y^(-1)=(20)/(y)`

Answer:

`20y^(-1)=(20)/(y)`