Answer :
To express the following rational numbers in standard form, follow these steps:
(i) [tex]\(\frac{-6}{15}\)[/tex]
- Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
- The GCD of 6 and 15 is 3.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-6}{15} = \frac{-6 \div 3}{15 \div 3} = \frac{-2}{5} \][/tex]
- Therefore, [tex]\(\frac{-6}{15}\)[/tex] in standard form is [tex]\(\frac{-2}{5}\)[/tex].
(ii) [tex]\(\frac{-18}{-24}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 18 and 24 is 6.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-18}{-24} = \frac{-18 \div 6}{-24 \div 6} = \frac{3}{4} \][/tex]
- Note that the signs cancel each other out, resulting in a positive fraction.
- Therefore, [tex]\(\frac{-18}{-24}\)[/tex] in standard form is [tex]\(\frac{3}{4}\)[/tex].
(iii) [tex]\(\frac{21}{-35}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 21 and 35 is 7.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{21}{-35} = \frac{21 \div 7}{-35 \div 7} = \frac{3}{-5} = -\frac{3}{5} \][/tex]
- The negative sign can be moved to the numerator.
- Therefore, [tex]\(\frac{21}{-35}\)[/tex] in standard form is [tex]\(\frac{-3}{5}\)[/tex].
(iv) [tex]\(\frac{-36}{84}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 36 and 84 is 12.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-36}{84} = \frac{-36 \div 12}{84 \div 12} = \frac{-3}{7} \][/tex]
- Therefore, [tex]\(\frac{-36}{84}\)[/tex] in standard form is [tex]\(\frac{-3}{7}\)[/tex].
(v) [tex]\(\frac{-1302}{-1953}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 1302 and 1953 is 651.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-1302}{-1953} = \frac{-1302 \div 651}{-1953 \div 651} = \frac{2}{3} \][/tex]
- Note that the signs cancel each other out, resulting in a positive fraction.
- Therefore, [tex]\(\frac{-1302}{-1953}\)[/tex] in standard form is [tex]\(\frac{2}{3}\)[/tex].
In summary, the rational numbers in standard form are:
(i) [tex]\(\frac{-2}{5}\)[/tex]
(ii) [tex]\(\frac{3}{4}\)[/tex]
(iii) [tex]\(\frac{-3}{5}\)[/tex]
(iv) [tex]\(\frac{-3}{7}\)[/tex]
(v) [tex]\(\frac{2}{3}\)[/tex]
(i) [tex]\(\frac{-6}{15}\)[/tex]
- Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
- The GCD of 6 and 15 is 3.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-6}{15} = \frac{-6 \div 3}{15 \div 3} = \frac{-2}{5} \][/tex]
- Therefore, [tex]\(\frac{-6}{15}\)[/tex] in standard form is [tex]\(\frac{-2}{5}\)[/tex].
(ii) [tex]\(\frac{-18}{-24}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 18 and 24 is 6.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-18}{-24} = \frac{-18 \div 6}{-24 \div 6} = \frac{3}{4} \][/tex]
- Note that the signs cancel each other out, resulting in a positive fraction.
- Therefore, [tex]\(\frac{-18}{-24}\)[/tex] in standard form is [tex]\(\frac{3}{4}\)[/tex].
(iii) [tex]\(\frac{21}{-35}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 21 and 35 is 7.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{21}{-35} = \frac{21 \div 7}{-35 \div 7} = \frac{3}{-5} = -\frac{3}{5} \][/tex]
- The negative sign can be moved to the numerator.
- Therefore, [tex]\(\frac{21}{-35}\)[/tex] in standard form is [tex]\(\frac{-3}{5}\)[/tex].
(iv) [tex]\(\frac{-36}{84}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 36 and 84 is 12.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-36}{84} = \frac{-36 \div 12}{84 \div 12} = \frac{-3}{7} \][/tex]
- Therefore, [tex]\(\frac{-36}{84}\)[/tex] in standard form is [tex]\(\frac{-3}{7}\)[/tex].
(v) [tex]\(\frac{-1302}{-1953}\)[/tex]
- Simplify the fraction by finding the GCD of the numerator and the denominator.
- The GCD of 1302 and 1953 is 651.
- Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{-1302}{-1953} = \frac{-1302 \div 651}{-1953 \div 651} = \frac{2}{3} \][/tex]
- Note that the signs cancel each other out, resulting in a positive fraction.
- Therefore, [tex]\(\frac{-1302}{-1953}\)[/tex] in standard form is [tex]\(\frac{2}{3}\)[/tex].
In summary, the rational numbers in standard form are:
(i) [tex]\(\frac{-2}{5}\)[/tex]
(ii) [tex]\(\frac{3}{4}\)[/tex]
(iii) [tex]\(\frac{-3}{5}\)[/tex]
(iv) [tex]\(\frac{-3}{7}\)[/tex]
(v) [tex]\(\frac{2}{3}\)[/tex]
