Question

Express the given rational number in the standard form \[\dfrac{{24}}{{ - 64}}\].

Answer 1

Hint: Here in this question we have determined the standard form of the \[\dfrac{{24}}{{ - 64}}\]. This can be determined by dividing both the numerator and denominator by a common number until we get the highest common factor of both numerator and denominator as 1. Hence we get a required result.

Complete step by step answer:
In mathematics we have different kinds of numbers namely, natural number, whole number, integers, rational number, irrational number and real number. The rational number is defined or represented in the form of \[\dfrac{p}{q}\] where \[q \ne 0\] and p , q can be integers.

Here we have to write the given rational number in its standard form. So the definition for the standard form of rational is defined as the standard form of a rational number: a rational number is said to be in standard form if the common factor between numerator and denominator is only 1 and the denominator is always positive. Now we write \[\dfrac{{24}}{{ - 64}}\] in the standard form.

Firstly we know the factors of 24 and 64.
The factors of 24 are \[1,2,3,4,6,8,12,24\].
The factors of 64 are \[1,2,4,8,16,32,64\].
Now we divide the both numerator and denominator by 2. On dividing we have
\[ \Rightarrow \dfrac{{12}}{{ - 32}}\]
Again on dividing both numerator and denominator by 2
\[ \Rightarrow \dfrac{6}{{ - 16}}\]
Again on dividing both numerator and denominator by 2
\[ \Rightarrow \dfrac{3}{{ - 8}}\]
Multiply both numerator and denominator by -1 we get
\[ \Rightarrow - \dfrac{3}{8}\]
Therefore the standard form of \[\dfrac{{24}}{{ - 64}}\] is \[ - \dfrac{3}{8}\]
Or else we can divide the both numerator and denominator of the number \[\dfrac{{24}}{{ - 64}}\] by 8. The number 8 is the HCF of the numbers 24 and 64
\[ \Rightarrow \dfrac{3}{{ - 8}}\]
Multiply both numerator and denominator by -1 we get
\[ \therefore - \dfrac{3}{8}\]

Hence, the rational number \[\dfrac{{24}}{{ - 64}}\] in the standard form is \[ -\dfrac{3}{8}\].

Note: Factors are whole numbers that are multiplied together to produce another number. The original numbers are factors of the product number. The HCF is abbreviated as Highest Common Factor. It is the highest number that is common in both factors of numbers.In the final answer we have a factor but there are no common factors between those two numbers. The tables of multiplication are very important.