Certainly! Let's address the problem step-by-step to check if a rational number is in its standard form, and if not, we'll convert it to its standard form.
### Step-by-Step Solution:
Definition:
A rational number [tex]\(\frac{a}{b}\)[/tex] is in its standard form if:
1. [tex]\(b > 0\)[/tex] (the denominator is a positive integer).
2. The greatest common divisor (gcd) of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is 1, meaning [tex]\(a\)[/tex] and [tex]\(b\)[/tex] have no common factors other than 1.
Given Rational Number:
[tex]\[
\frac{8}{12}
\][/tex]
1. Check if the denominator is positive:
The given denominator is 12, which is a positive integer. Thus, the first condition is already satisfied.
2. Find the gcd of the numerator and the denominator:
We calculate the greatest common divisor (gcd) of 8 and 12.
- The factors of 8 are: 1, 2, 4, 8
- The factors of 12 are: 1, 2, 3, 4, 6, 12
- The common factors of 8 and 12 are: 1, 2, 4
The greatest of these common factors is 4. So, [tex]\(\text{gcd}(8, 12) = 4\)[/tex].
3. Check if gcd is 1:
Since the gcd of 8 and 12 is 4 (which is not 1), the rational number [tex]\(\frac{8}{12}\)[/tex] is not in its standard form.
4. Convert the rational number to its standard form:
To simplify the fraction [tex]\(\frac{8}{12}\)[/tex], we divide both the numerator and the denominator by their gcd (which is 4):
[tex]\[
\text{Simplified numerator} = \frac{8}{4} = 2
\][/tex]
[tex]\[
\text{Simplified denominator} = \frac{12}{4} = 3
\][/tex]
Thus, the simplified form of the rational number [tex]\(\frac{8}{12}\)[/tex] produces the fraction [tex]\(\frac{2}{3}\)[/tex].
### Conclusion:
1. The rational number [tex]\(\frac{8}{12}\)[/tex] is not in its standard form.
2. When simplified, the standard form of [tex]\(\frac{8}{12}\)[/tex] is [tex]\(\frac{2}{3}\)[/tex].
