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Sagot :
Certainly! Let's address each part of the problem step by step with clear explanations.
### Part 6: Write Rational Numbers
A rational number is defined as a number that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex] of two integers, where [tex]\(a\)[/tex] (the numerator) and [tex]\(b\)[/tex] (the denominator), and [tex]\(b \neq 0\)[/tex].
(a) Rational number with numerator [tex]\(-3\)[/tex] and denominator [tex]\(4\)[/tex]:
[tex]\[ \frac{-3}{4} = -0.75 \][/tex]
(b) Rational number with numerator [tex]\(-16\)[/tex] and denominator [tex]\(-32\)[/tex]:
[tex]\[ \frac{-16}{-32} = \frac{16}{32} = 0.5 \][/tex]
Since the negative signs cancel each other out, this simplifies to [tex]\(\frac{1}{2}\)[/tex], which is [tex]\(0.5\)[/tex].
(c) Rational number with numerator [tex]\(18 + 6\)[/tex] and denominator [tex]\(4 - 17\)[/tex]:
[tex]\[ \frac{18 + 6}{4 - 17} = \frac{24}{-13} = -1.8461538461538463 \][/tex]
The numerator simplifies to [tex]\(24\)[/tex] and the denominator simplifies to [tex]\(-13\)[/tex].
### Part 7: Determine Whether the Given Numbers are Rational
A number is rational if it can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
(d) [tex]\((-6)^2\)[/tex]:
[tex]\[ (-6)^2 = 36 \][/tex]
This is a perfect square and can be written as the fraction [tex]\(\frac{36}{1}\)[/tex]. Hence, [tex]\(36\)[/tex] is a rational number.
(e) [tex]\((-15) \div 3\)[/tex]:
[tex]\[ \frac{-15}{3} = -5.0 \][/tex]
This division results in [tex]\(-5.0\)[/tex], which can be expressed as [tex]\(\frac{-5}{1}\)[/tex]. Hence, [tex]\(-5.0\)[/tex] is a rational number.
### Summary
1. The rational numbers for part 6 are:
- [tex]\( \frac{-3}{4} = -0.75 \)[/tex]
- [tex]\( \frac{-16}{-32} = 0.5 \)[/tex]
- [tex]\( \frac{24}{-13} = -1.8461538461538463 \)[/tex]
2. Both numbers in part 7 are rational:
- [tex]\( (-6)^2 = 36 \)[/tex]
- [tex]\( (-15) \div 3 = -5.0 \)[/tex]
By expressing these results as fractions where applicable, we confirm their rationality.
### Part 6: Write Rational Numbers
A rational number is defined as a number that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex] of two integers, where [tex]\(a\)[/tex] (the numerator) and [tex]\(b\)[/tex] (the denominator), and [tex]\(b \neq 0\)[/tex].
(a) Rational number with numerator [tex]\(-3\)[/tex] and denominator [tex]\(4\)[/tex]:
[tex]\[ \frac{-3}{4} = -0.75 \][/tex]
(b) Rational number with numerator [tex]\(-16\)[/tex] and denominator [tex]\(-32\)[/tex]:
[tex]\[ \frac{-16}{-32} = \frac{16}{32} = 0.5 \][/tex]
Since the negative signs cancel each other out, this simplifies to [tex]\(\frac{1}{2}\)[/tex], which is [tex]\(0.5\)[/tex].
(c) Rational number with numerator [tex]\(18 + 6\)[/tex] and denominator [tex]\(4 - 17\)[/tex]:
[tex]\[ \frac{18 + 6}{4 - 17} = \frac{24}{-13} = -1.8461538461538463 \][/tex]
The numerator simplifies to [tex]\(24\)[/tex] and the denominator simplifies to [tex]\(-13\)[/tex].
### Part 7: Determine Whether the Given Numbers are Rational
A number is rational if it can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
(d) [tex]\((-6)^2\)[/tex]:
[tex]\[ (-6)^2 = 36 \][/tex]
This is a perfect square and can be written as the fraction [tex]\(\frac{36}{1}\)[/tex]. Hence, [tex]\(36\)[/tex] is a rational number.
(e) [tex]\((-15) \div 3\)[/tex]:
[tex]\[ \frac{-15}{3} = -5.0 \][/tex]
This division results in [tex]\(-5.0\)[/tex], which can be expressed as [tex]\(\frac{-5}{1}\)[/tex]. Hence, [tex]\(-5.0\)[/tex] is a rational number.
### Summary
1. The rational numbers for part 6 are:
- [tex]\( \frac{-3}{4} = -0.75 \)[/tex]
- [tex]\( \frac{-16}{-32} = 0.5 \)[/tex]
- [tex]\( \frac{24}{-13} = -1.8461538461538463 \)[/tex]
2. Both numbers in part 7 are rational:
- [tex]\( (-6)^2 = 36 \)[/tex]
- [tex]\( (-15) \div 3 = -5.0 \)[/tex]
By expressing these results as fractions where applicable, we confirm their rationality.
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