Question

Find the square root of 9216 and 1024 and hence find the value of √0.9216 + √0.1024

Answer 1

$\Large {\underline { \sf {Explication \: of \: steps :}}}$

$\underline{ \sf {\maltese \; \; \; Finding \: square \: root \: of \: 9216 : \; \; \; }}$

By prime factorization,

$\begin{array}{c|c} \underline{\sf {6}}&\underline{\sf {\; \; 9216 \; \; \: }} \\ \underline{\sf {6}}&\underline{\sf {\; \; 1536 \; \; \: }}\\ \underline{\sf {4}}&\underline{\sf {\; \; \: 256 \: \; \; \: }} \\ \underline{\sf {4}}&\underline{\sf {\; \; \: \: 64 \: \: \; \; \: }} \\\underline{\sf {4}}&\underline{\sf {\; \; \: 16 \: \; \; \: }} \\\underline{\sf {4}}&\underline{\sf {\; \; \: \: \: 4 \: \: \: \; \; \: }} \\& \sf {\; \; \: 1 \: \; \; }\end{array}$

→ 9216 = 6 × 6 × 4 × 4 × 4 × 4

• Making pairs and taking out one common factor from each pair. Product of those factors will be the square root of 9216.

→ 9216 = 6 × 6 × 4 × 4 × 4 × 4

→ √9216 = 6 × 4 × 4

→ √9216 = 96

$\underline{ \sf {\maltese \; \; \; Finding \: square \: root \: of \: 1024 : \; \; \; }}$

By prime factorization,

$\begin{array}{c|c} \underline{\sf {2}}&\underline{\sf {\; \; 1024 \; \; \: }} \\ \underline{\sf {2}}&\underline{\sf {\; \; 512 \; \; \: }}\\ \underline{\sf {4}}&\underline{\sf {\; \; \: 256 \: \; \; \: }} \\ \underline{\sf {4}}&\underline{\sf {\; \; \: \: 64 \: \: \; \; \: }} \\\underline{\sf {4}}&\underline{\sf {\; \; \: 16 \: \; \; \: }} \\\underline{\sf {4}}&\underline{\sf {\; \; \: \: \: 4 \: \: \: \; \; \: }} \\& \sf {\; \; \: 1 \: \; \; }\end{array}$

→ 1024 = 2 × 2 × 4 × 4 × 4 × 4

• Making pairs and taking out one common factor from each pair. Product of those factors will be the square root of 9216.

→ 1024 = 2 × 2 × 4 × 4 × 4 × 4

→ √1024 = 2 × 4 × 4

→ √1024 = 32

_________________

$\underline{ \sf {\maltese \; \; \; Finding \: the \: value \: of \: \sqrt{0.9216} + \sqrt{0.1024} : \; \; \; }}$

→ √0.9216 + √0.1024

→ $\sf {\sqrt { \dfrac{9216}{10000} } + \sqrt{ \dfrac{1024}{10000} }}$

→ $\sf { \dfrac{\sqrt{9216}}{\sqrt{10000}} + \dfrac{\sqrt{1024}}{\sqrt{1000}} }$

• Square root of 10000 is 100.

By prime factorization,

$\begin{array}{c|c} \underline{\sf {10}}&\underline{\sf {\; \; 10000 \; \; \: }} \\ \underline{\sf {10}}&\underline{\sf {\; \; 1000 \; \; \: }}\\ \underline{\sf {10}}&\underline{\sf {\; \; \: 100 \: \; \; \: }} \\ \underline{\sf {10}}&\underline{\sf {\; \; \: \: 10 \: \: \; \; \: }} \\& \sf {\; \; \: 1 \: \; \; }\end{array}$

⇒ 10000 = 10 × 10 × 10 × 10

⇒ 10000 = 10 × 10 × 10 × 10

⇒ √10000 = 10 × 10

⇒ √10000 = 100

→ $\sf {\dfrac{96}{100} + \dfrac{32}{100} }$

→ $\sf {\dfrac{96+32}{100}}$

→ $\sf {\dfrac{128}{100}}$

→ 1.28

So, we get that ,

$\mapsto {\underline {\boxed {\sf { \sqrt{0.9216} + \sqrt{0.1024} = 1.28 } }}}$