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Which expressions represent rational numbers? Select all that apply.

A. [tex]\sqrt{100} \sqrt{100}[/tex]
B. [tex]13.5+\sqrt{81}[/tex]
C. [tex]\sqrt{9}+\sqrt{729}[/tex]
D. [tex]\sqrt{64}+\sqrt{353}[/tex]
E. [tex]\frac{1}{3}+\sqrt{216}[/tex]
F. [tex]\frac{3}{5}+2.5[/tex]


Sagot :

To determine which expressions represent rational numbers, we need to simplify each expression and check if its components are rational numbers. A rational number is any number that can be expressed as the quotient of two integers (i.e., as a fraction), including terminating decimals.

1. [tex]\(\sqrt{100} + \sqrt{100}\)[/tex]:
[tex]\[ \sqrt{100} = 10 \][/tex]
[tex]\[ \sqrt{100} + \sqrt{100} = 10 + 10 = 20 \][/tex]
Since 20 is a rational number, this expression is rational.

2. [tex]\(13.5 + \sqrt{81}\)[/tex]:
[tex]\[ \sqrt{81} = 9 \][/tex]
[tex]\[ 13.5 + 9 = 22.5 \][/tex]
Since 22.5 is a rational number, this expression is rational.

3. [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]:
[tex]\[ \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{729} = 27 \][/tex]
[tex]\[ \sqrt{9} + \sqrt{729} = 3 + 27 = 30 \][/tex]
Since 30 is a rational number, this expression is rational.

4. [tex]\(\sqrt{64} + \sqrt{353}\)[/tex]:
[tex]\[ \sqrt{64} = 8 \][/tex]
[tex]\(\sqrt{353}\)[/tex] is non-integer because 353 is not a perfect square.
[tex]\[ \sqrt{64} + \sqrt{353} = 8 + \sqrt{353} \][/tex]
Since [tex]\(\sqrt{353}\)[/tex] is an irrational number, this expression is not rational.

5. [tex]\(\frac{1}{3} + \sqrt{216}\)[/tex]:
[tex]\[ \frac{1}{3} = 0.333\ldots \][/tex]
[tex]\(\sqrt{216}\)[/tex] is non-integer because 216 is not a perfect square.
[tex]\[ \frac{1}{3} + \sqrt{216} \][/tex]
Since [tex]\(\sqrt{216}\)[/tex] is an irrational number, this expression is not rational.

6. [tex]\(\frac{3}{5} + 2.5\)[/tex]:
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
[tex]\[ 2.5 = 2.5 \text{ (rational number since it is a terminating decimal)} \][/tex]
[tex]\[ \frac{3}{5} + 2.5 = 0.6 + 2.5 = 3.1 \][/tex]
Since 3.1 is a rational number, this expression is rational.

Summarizing the results:

- [tex]\(\sqrt{100} + \sqrt{100}\)[/tex] is rational.
- [tex]\(13.5 + \sqrt{81}\)[/tex] is rational.
- [tex]\(\sqrt{9} + \sqrt{729}\)[/tex] is rational.
- [tex]\(\sqrt{64} + \sqrt{353}\)[/tex] is not rational.
- [tex]\(\frac{1}{3} + \sqrt{216}\)[/tex] is not rational.
- [tex]\(\frac{3}{5} + 2.5\)[/tex] is rational.

Therefore, the expressions that represent rational numbers are:
- [tex]\(\sqrt{100} + \sqrt{100}\)[/tex]
- [tex]\(13.5 + \sqrt{81}\)[/tex]
- [tex]\(\sqrt{9} + \sqrt{729}\)[/tex]
- [tex]\(\frac{3}{5} + 2.5\)[/tex]