Question

Simplify each of the following and express the result as a rational number in standard form:
\[\dfrac{{ - 16}}{{21}} \times \dfrac{{14}}{5}\].

Answer 1

Hint: To solve this question, we will first simplify both the terms separately. We will then multiply the terms and express them as a rational number. Rational number is a fraction (or \[\dfrac{p}{q}\] form) where numerator and denominator are integers and the denominator is not equal to zero.

Complete step-by-step answer:
We will first simplify \[\dfrac{{ - 16}}{{21}}\].
Now \[\dfrac{{ - 16}}{{21}}\] is already in its lowest form and cannot be simplified further. This is because the numerator is divisible by 2 but not by 3. The denominator is divisible by 3 but not by 2. Hence, they have no common factors. So, we can say that it is already in its lowest form.
Now we will simplify \[\dfrac{{14}}{5}\].
\[\dfrac{{14}}{5}\] is already in its lowest form and cannot be simplified further. This is because the numerator is divisible by 2 but not by 5. The denominator is divisible by 5 but not by 2. Hence, they have no common factors. So, we can say that it is already in its lowest form.
Now as both the terms are simplified so we will multiply the terms now.
We know that when the rational numbers are multiplied with each other, then the numerators are multiplied together and the denominators are multiplied together. So, on multiplying the terms we get,
\[\dfrac{{ - 16}}{{21}} \times \dfrac{{14}}{5}\]
We can see that the numerator of \[\dfrac{{14}}{5}\]i.e., 14 and the denominator of \[\dfrac{{ - 16}}{{21}}\],i.e., 21 have the common factor as 7. Hence, on dividing both of these terms by 7, we get
\[\dfrac{{ - 16}}{{21}} \times \dfrac{{14}}{5} = \dfrac{{ - 16}}{3} \times \dfrac{2}{5}\]
Now on multiplying the terms, we get
\[\dfrac{{ - 16}}{3} \times \dfrac{2}{5} = \dfrac{{ - 32}}{{15}}\]
Now we know that \[\dfrac{{ - 32}}{{15}}\] is already in its lowest form and cannot be simplified further.
The fraction obtained is in the form of a fraction \[\dfrac{p}{q}\], where \[q \ne 0\]. Therefore, this is the required rational number in the standard form.

Note: The difference between fractions and rational numbers, however, is that rational numbers are both positive and negative, unlike the fractions. Also, when the denominator is 1, rational numbers become integers. Thus, it is said that all integers are rational numbers, but not all rational numbers are integers.