Asked by rod wave
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
π+4–√
pi plus Start Root 4 End Root
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.6¯¯¯¯+910
Responses
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
π+4–√
pi plus Start Root 4 End Root
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.6¯¯¯¯+910
Answers
Answer
In the proof that ab+cd
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses
By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses
By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
Answer
The sum of two rational numbers is always(1 point)
Responses
zero.
zero.
irrational.
irrational.
rational.
rational.
radical.
Responses
zero.
zero.
irrational.
irrational.
rational.
rational.
radical.
Answered by
GPT-4o mini
To determine which of the sums results in a rational number, we need to evaluate each expression:
1. **\(0.643892553... + 34\)**: The number \(0.643892553...\) is a non-repeating decimal and thus an irrational number. The sum of an irrational number and a rational number (34) is irrational, so this expression does not yield a rational number.
2. **\(\pi + 4 - \sqrt{4}\)**: Here, \(\pi\) is an irrational number, \(4\) is rational, and \(\sqrt{4} = 2\) is rational. Therefore, the expression simplifies to \(\pi + 4 - 2 = \pi + 2\), which is still irrational because the sum of an irrational number and a rational number is irrational. Thus, this does not yield a rational number.
3. **\(\sqrt{36} + \sqrt{221}\)**: The square root of \(36\) is \(6\), which is rational, but \(\sqrt{221}\) is irrational (since \(221\) is not a perfect square). The sum \(6 + \sqrt{221}\) will be irrational (as the sum of a rational and an irrational number is irrational). So this does not yield a rational number.
4. **\(0.6\overline{6} + \frac{9}{10}\)**: The number \(0.6\overline{6}\) is a repeating decimal, which is rational (it can be expressed as \( \frac{2}{3} \)). The fraction \(\frac{9}{10}\) is also rational. The sum of two rational numbers is rational. Thus, \(0.6\overline{6} + \frac{9}{10}\) results in a rational number.
Therefore, the correct answer is:
**\(0.6\overline{6} + \frac{9}{10}\)** results in a rational number.
1. **\(0.643892553... + 34\)**: The number \(0.643892553...\) is a non-repeating decimal and thus an irrational number. The sum of an irrational number and a rational number (34) is irrational, so this expression does not yield a rational number.
2. **\(\pi + 4 - \sqrt{4}\)**: Here, \(\pi\) is an irrational number, \(4\) is rational, and \(\sqrt{4} = 2\) is rational. Therefore, the expression simplifies to \(\pi + 4 - 2 = \pi + 2\), which is still irrational because the sum of an irrational number and a rational number is irrational. Thus, this does not yield a rational number.
3. **\(\sqrt{36} + \sqrt{221}\)**: The square root of \(36\) is \(6\), which is rational, but \(\sqrt{221}\) is irrational (since \(221\) is not a perfect square). The sum \(6 + \sqrt{221}\) will be irrational (as the sum of a rational and an irrational number is irrational). So this does not yield a rational number.
4. **\(0.6\overline{6} + \frac{9}{10}\)**: The number \(0.6\overline{6}\) is a repeating decimal, which is rational (it can be expressed as \( \frac{2}{3} \)). The fraction \(\frac{9}{10}\) is also rational. The sum of two rational numbers is rational. Thus, \(0.6\overline{6} + \frac{9}{10}\) results in a rational number.
Therefore, the correct answer is:
**\(0.6\overline{6} + \frac{9}{10}\)** results in a rational number.
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