According to the Closure Property, which of these sums results in a rational number?(1 point)

Question

Responses

36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root

0.643892553…+3/4
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction

0.6¯¯¯¯+9/10
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction

π+4–√

In the proof that a/b+c/d
, 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/bd
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+bc/bd
is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both quotients of 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, ad+bc
and bd
are both integers, and so ad+bc/bd
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.

Answer 1

To answer your first question about which of the sums results in a rational number:

1. \( \sqrt{36} + \sqrt{221} \): \( \sqrt{36} = 6 \) (a rational number), but \( \sqrt{221} \) is not a rational number (it cannot be expressed as a fraction), so their sum is not rational.

2. \( 0.643892553... + \frac{3}{4} \): The first number is a decimal that is not terminating or repeating, hence not rational. The second number is rational, but the sum of a rational and an irrational number is irrational.

3. \( 0.6\overline{6} + \frac{9}{10} \): \( 0.6\overline{6} \) is a repeating decimal which is rational (equal to \( \frac{2}{3} \)). The sum with \( \frac{9}{10} \) results in \( \frac{2}{3} + \frac{9}{10} = \frac{20}{30} + \frac{27}{30} = \frac{47}{30} \), which is rational.

4. \( \pi + \sqrt{4} \): \( \sqrt{4} = 2 \) (a rational number), but \( \pi \) is irrational, so their sum is irrational.

Conclusion: The sum \( 0.6\overline{6} + \frac{9}{10} \) results in a rational number.

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For your second question regarding why \( \frac{ad + bc}{bd} \) is a rational number:

The correct response is: By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.

This explains that since both the numerator and denominator are integers (as a product of integers is also an integer), the overall fraction represents a rational number.