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Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:
\( \sqrt{\frac{9}{27}} \)
Given:
\( \sqrt{\frac{9}{27}} \)
To do:
We have to identify the given number as rational or irrational and write its decimal representation.
Solution:
A rational number can be expressed in either terminating decimal or non-terminating recurring decimals and an irrational number is expressed in non-terminating non-recurring decimals.
Therefore,
$\sqrt{\frac{9}{27}}=\sqrt{\frac{1}{3}}$
$=\frac{1}{\sqrt{3}}$
$=\frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}$
$=\frac{\sqrt{3}}{3}$
$=\frac{1.7320508075.....}{3}$
$=0.5773502692.......$
The decimal expansion of \( \sqrt{\frac{9}{27}} \) is $0.5773502692.......$ and it is non-terminating non-recurring.
Therefore, \( \sqrt{\frac{9}{27}} \) is an irrational number.