Question

Simplyify:√900+√0.09+√0.000009

Answer 1

$\sf{ \sqrt{900} + \sqrt{0.09} + \sqrt{0.000009} }$

Let us first convert & write the decimals into fraction!

$\sf{ \longrightarrow \: \sqrt{900} + \sqrt{ \dfrac{9}{100} } + \sqrt{ \dfrac{9}{1000000}} }$

$\sf{ \longrightarrow \: \sqrt{900} + \dfrac{ \sqrt{9}}{ \sqrt{100}} + \dfrac{ \sqrt{9}} { \sqrt{ 1000000}} }$

Since, $\sf { \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}}$

Now, finding square roots by prime factorization:

$\rm\purple { Square \: root \: of \: 900 :}$

$\large {\begin{array}{c|c} \large \underline{3} & \underline{900} \\ \underline{3} & \underline{300} \\ \underline{10} & \underline{100} \\ \underline{10} & \underline{10} \\ \: & 1 \end{array} }$

→ 900 = 3 × 3 × 10 × 10

→ 900 = 3² × 10²

→ √900 = √(3² × 10²)

→ √900 = √(3²) × √(10²)

→ √900 = 3 × 10

→ √900 = 30

$\rm\purple { Square \: root \: of \: 9 :}$

$\large {\begin{array}{c|c} \large \underline{3} & \underline{9} \\ \underline{3} & \underline{3} \\ \: & 1 \end{array} }$

→ 9 = 3 × 3

→ 9 = 3²

→ √9 = √3²

→ √9 = 3

$\rm\purple { Square \: root \: of \: 100 :}$

$\large {\begin{array}{c|c} \large \underline{10} & \underline{100} \\ \underline{10} & \underline{10} \\ \: & 1 \end{array} }$

→ 100 = 10 × 10

→ 100 = 10²

→ √100 = √10²

→ √100 = 10

$\rm\purple { Square \: root \: of \: 1000000 :}$

$\large {\begin{array}{c|c} \large \underline{10} & \underline{1000000} \\ \underline{10} & \underline{100000} \\ \underline{10} & \underline{10000} \\ \underline{10} & \underline{1000} \\ \underline{10} & \underline{100} \\\underline{10} & \underline{10} \\ \: & 1 \end{array} }$

→ 1000000 = 10 × 10 × 10 × 10 × 10 × 10

→ √1000000 = 10 × 10 × 10

→ √1000000 = 1000

Substituting all the square roots :

$\sf{ \longrightarrow \: 30+ \dfrac{ 3}{10} + \dfrac{3} { 1000} } \\ \\ \sf{ \longrightarrow \: \dfrac{30000 + 300 + 3} { 1000} } \\ \\ \sf{ \longrightarrow \: \dfrac{30303 } { 1000} } \\ \\ \longrightarrow\underline{\boxed{\sf{30.303}}} \: \red{\bigstar}$

Henceforth, 30.303 is the answer.

Answer 2

√(900) + √(0.09) + √(0.000009)

= √(3² × 10²) + √(3² × 10^-2) + √(3² × 10^-6)

= 3 × 10 + 3 × 10^-1 + 3 × 10^-3

= 30 + 0.3 + 0.003

= 30.3003 (answer).