Question

square root - perfect square
a) find the values of the following.

1. $sqrt{\frac{4}{36}}$ 2) $sqrt{\frac{169}{81}}$ 3) $sqrt{\frac{676}{9}}$ 4) $sqrt{\frac{361}{64}}$ 5) $sqrt{\frac{25}{100}}$ 6) $sqrt{\frac{400}{225}}$

b) find the square roots of the following numerals.

1. $\frac{529}{841}$ 2) $\frac{4}{16}$ 3) $\frac{784}{625}$ 4) $\frac{961}{36}$ 5) $\frac{441}{289}$ 6) $\frac{196}{784}$

c) 1) find the value of $sqrt{\frac{4}{9}}$.
i) $\frac{2}{3}$ ii) $\frac{3}{4}$ iii) $\frac{4}{3}$ iv) $\frac{8}{18}$

1. find the number, when multiplied by itself gives $\frac{144}{361}$.

i) $\frac{6}{3}$ ii) $\frac{12}{19}$ iii) $\frac{19}{12}$ iv) $\frac{18}{19}$

Explanation:

Step1: Recall square - root property

For $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ where $a\geq0$ and $b > 0$.

Step2: Solve part A - 1

For $\sqrt{\frac{4}{36}}$, we know that $\sqrt{4} = 2$ and $\sqrt{36}=6$, so $\sqrt{\frac{4}{36}}=\frac{\sqrt{4}}{\sqrt{36}}=\frac{2}{6}=\frac{1}{3}$.

Step3: Solve part A - 2

For $\sqrt{\frac{169}{81}}$, since $\sqrt{169} = 13$ and $\sqrt{81}=9$, then $\sqrt{\frac{169}{81}}=\frac{13}{9}$.

Step4: Solve part A - 3

For $\sqrt{\frac{676}{9}}$, as $\sqrt{676}=26$ and $\sqrt{9} = 3$, so $\sqrt{\frac{676}{9}}=\frac{26}{3}$.

Step5: Solve part A - 4

For $\sqrt{\frac{361}{64}}$, because $\sqrt{361}=19$ and $\sqrt{64}=8$, then $\sqrt{\frac{361}{64}}=\frac{19}{8}$.

Step6: Solve part A - 5

For $\sqrt{\frac{25}{100}}$, since $\sqrt{25}=5$ and $\sqrt{100}=10$, so $\sqrt{\frac{25}{100}}=\frac{5}{10}=\frac{1}{2}$.

Step7: Solve part A - 6

For $\sqrt{\frac{400}{225}}$, as $\sqrt{400}=20$ and $\sqrt{225}=15$, then $\sqrt{\frac{400}{225}}=\frac{20}{15}=\frac{4}{3}$.

Step8: Solve part B - 1

For $\sqrt{\frac{529}{841}}$, since $529 = 23\times23$ and $841=29\times29$, then $\sqrt{\frac{529}{841}}=\frac{23}{29}$.

Step9: Solve part B - 2

For $\sqrt{\frac{4}{16}}$, we have $\sqrt{4} = 2$ and $\sqrt{16}=4$, so $\sqrt{\frac{4}{16}}=\frac{2}{4}=\frac{1}{2}$.

Step10: Solve part B - 3

For $\sqrt{\frac{784}{625}}$, as $784 = 28\times28$ and $625=25\times25$, then $\sqrt{\frac{784}{625}}=\frac{28}{25}$.

Step11: Solve part B - 4

For $\sqrt{\frac{961}{36}}$, since $961 = 31\times31$ and $\sqrt{36}=6$, so $\sqrt{\frac{961}{36}}=\frac{31}{6}$.

Step12: Solve part B - 5

For $\sqrt{\frac{441}{289}}$, as $441 = 21\times21$ and $289=17\times17$, then $\sqrt{\frac{441}{289}}=\frac{21}{17}$.

Step13: Solve part B - 6

For $\sqrt{\frac{196}{784}}$, we know that $\sqrt{196}=14$ and $\sqrt{784}=28$, so $\sqrt{\frac{196}{784}}=\frac{14}{28}=\frac{1}{2}$.

Step14: Solve part C - 1

For $\sqrt{\frac{4}{9}}$, since $\sqrt{4} = 2$ and $\sqrt{9}=3$, the value is $\frac{2}{3}$, so the answer is i).

Step15: Solve part C - 2

Let the number be $x$. We want $x^2=\frac{144}{361}$, then $x=\sqrt{\frac{144}{361}}$. Since $\sqrt{144}=12$ and $\sqrt{361}=19$, $x = \frac{12}{19}$, so the answer is ii).

Answer:

A - 1: $\frac{1}{3}$
A - 2: $\frac{13}{9}$
A - 3: $\frac{26}{3}$
A - 4: $\frac{19}{8}$
A - 5: $\frac{1}{2}$
A - 6: $\frac{4}{3}$
B - 1: $\frac{23}{29}$
B - 2: $\frac{1}{2}$
B - 3: $\frac{28}{25}$
B - 4: $\frac{31}{6}$
B - 5: $\frac{21}{17}$
B - 6: $\frac{1}{2}$
C - 1: i) $\frac{2}{3}$
C - 2: ii) $\frac{12}{19}$