Question

1. circle any number below that is a square number (the root is a repeating or terminating decimal), cross off any number that is not. (5 marks)

a) 12.1
b) 1.44
c) \\(\frac{9}{25}\\)
d) \\(\frac{17}{25}\\)
e) \\(\frac{36}{121}\\)

Explanation:

Analyze the definition of a square number

To determine if a number is a square number, we find its square root. The problem specifies that the square root must be a repeating or terminating decimal (which means the square root must be a rational number).

Evaluate part a

Using the Square Root of Decimals and Perfect Square Decimals knowledge points

$$ \sqrt{12.1} = \sqrt{\frac{121}{10}} = \frac{11}{\sqrt{10}} \approx 3.4785 $$

Since \(\sqrt{10}\) is irrational, the square root is an infinite, non-repeating decimal. Thus, \(12.1\) is not a square number.

Evaluate part b

Using the Square Root of Decimals and Perfect Square Decimals knowledge points

$$ \sqrt{1.44} = \sqrt{\frac{144}{100}} = \frac{12}{10} = 1.2 $$

Since \(1.2\) is a terminating decimal, \(1.44\) is a square number.

Evaluate part c

Using the Square Root of Fractions and Perfect Square Fractions knowledge points

$$ \sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5} = 0.6 $$

Since \(0.6\) is a terminating decimal, \(\frac{9}{25}\) is a square number.

Evaluate part d

Using the Square Root of Fractions and Perfect Square Fractions knowledge points

$$ \sqrt{\frac{17}{25}} = \frac{\sqrt{17}}{\sqrt{25}} = \frac{\sqrt{17}}{5} \approx 0.8246 $$

Since \(\sqrt{17}\) is irrational, the square root is an infinite, non-repeating decimal. Thus, \(\frac{17}{25}\) is not a square number.

Evaluate part e

Using the Square Root of Fractions and Perfect Square Fractions knowledge points

$$ \sqrt{\frac{36}{121}} = \frac{\sqrt{36}}{\sqrt{121}} = \frac{6}{11} = 0.\overline{54} $$

Since \(0.\overline{54}\) is a repeating decimal, \(\frac{36}{121}\) is a square number.

Answer:

Question a

Cross off: \(12.1\) is not a square number.

Question b

Circle: \(1.44\) is a square number because \(\sqrt{1.44} = 1.2\).

Question c

Circle: \(\frac{9}{25}\) is a square number because \(\sqrt{\frac{9}{25}} = \frac{3}{5} = 0.6\).

Question d

Cross off: \(\frac{17}{25}\) is not a square number.

Question e

Circle: \(\frac{36}{121}\) is a square number because \(\sqrt{\frac{36}{121}} = \frac{6}{11} = 0.\overline{54}\).