Question

a) find the values of the following.

1. $sqrt{\frac{4}{576}}$ 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 $b > 0$. Also, if $x^2 = y$, then $x=\pm\sqrt{y}$. We consider the principal square - root (non - negative) for the given problems.

Step2: Solve part A - 1

For $\sqrt{\frac{4}{576}}$, we know that $\sqrt{4} = 2$ and $\sqrt{576}=24$, so $\sqrt{\frac{4}{576}}=\frac{2}{24}=\frac{1}{12}$.

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 $\sqrt{9}=3$, we have $\sqrt{\frac{676}{9}}=\frac{26}{3}$.

Step5: Solve part A - 4

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

Step6: Solve part A - 5

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

Step7: Solve part A - 6

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

Step8: Solve part B - 1

For $\sqrt{\frac{529}{841}}$, since $529 = 23^2$ and $841=29^2$, then $\sqrt{\frac{529}{841}}=\frac{23}{29}$.

Step9: Solve part B - 2

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

Step10: Solve part B - 3

For $\sqrt{\frac{784}{625}}$, because $\sqrt{784} = 28$ and $\sqrt{625}=25$, we have $\sqrt{\frac{784}{625}}=\frac{28}{25}$.

Step11: Solve part B - 4

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

Step12: Solve part B - 5

For $\sqrt{\frac{441}{289}}$, as $\sqrt{441} = 21$ and $\sqrt{289}=17$, we get $\sqrt{\frac{441}{289}}=\frac{21}{17}$.

Step13: Solve part B - 6

For $\sqrt{\frac{196}{784}}$, since $\sqrt{196} = 14$ and $\sqrt{784}=28$, then $\sqrt{\frac{196}{784}}=\frac{14}{28}=\frac{1}{2}$.

Step14: Solve part C - 1

For $\sqrt{\frac{4}{9}}$, as $\sqrt{4} = 2$ and $\sqrt{9}=3$, the value is $\frac{2}{3}$.

Step15: Solve part C - 2

Let the number be $x$. We have $x^2=\frac{144}{361}$. Then $x=\sqrt{\frac{144}{361}}$. Since $\sqrt{144} = 12$ and $\sqrt{361}=19$, $x = \frac{12}{19}$.

Answer:

A - 1: $\frac{1}{12}$
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: $\frac{2}{3}$
C - 2: $\frac{12}{19}$