Question

1 ) Find the square root of : 900, 7056.

2 ) By what smallest 180 must be multiplied so that the product becomes a perfect square? What is the square root of the number thus obtained?​

Answer 1

Solution :

Q 1)

• [i] 900

Express 900 into prime factors :

$\begin{array}{r | l}\underline2 & \underline{900} \\ \underline2 & \underline{450} \\ \underline 3 &\underline{ 225} \\ \underline3 & \underline{75} \\ \underline5 & \underline{25 }\\& {5} \\ \end{array}$

We get,

• 900 = 2×2 × 3×3 × 5×5 = 2² × 3² × 5²
• = ( 2 × 3 × 5 )²

$\therefore \: \: \sqrt{900} = 2 \times3 \times5 = \bf30$

• [ii] 7056

Express 7056 into prime factors :

$\begin{array}{r | l} \underline2 & \underline{7056} \\ \underline2 & \underline{3528} \\ \underline 2 &\underline{ 1764} \\ \underline2 & \underline{882} \\ \underline3 & \underline{441 }\\ \underline3& \underline{147} \\ \underline7& \underline{49} \\ & {7} \\ \end{array}$

We get,

• 7056 = 2×2 × 2×2 × 3×3 × 7×7 = 2² × 2² × 3² × 7²
• = ( 2 × 2 × 3 × 7 )²

$\therefore \: \: \sqrt{7056} = 2 \times2 \times3 \times7 = \bf84$

Q 2)

• 180 = 9 × 4 × 5
• = 3² × 2² × 5

In the right hand side, 5 is alone.

$\therefore \: 5 \: \: \sf{must \: be \: multiplied \: by} \\ \sf{another \: 5 \: to \: make \: it \: a \: square.}$

Multiply both sides by 5 :

$\therefore \: 180 \times5 = {3}^{2} \times {2}^{2} \times5 \times5 \\ = {3}^{2} \times {2}^{2} \times {5}^{2} \\ \\ \therefore \: \tt{Multiplying \: 180 \: by \: 5} \\ \tt \: we \: have \: a \: perfect \: square \: number \\ \tt \: and \: the \: number \: is \: \bf{900}$

By Prime factorisation :

$\begin{array}{r | l} \underline2 & \underline{900} \\ \underline2 & \underline{450} \\ \underline 3 &\underline{ 225} \\ \underline3 & \underline{75} \\ \underline5 & \underline{25 }\\ & {5} \\ \end{array}$

We get,

The square root of 900 :

$\therefore \: \: \sqrt{900} = 2 \times3 \times5 = \bf30$

Answer 2

Here are the answers:-

1) Pls find in attachment.

2)Square Root of 180: √180 = 13.416.

Square of 180: 1802 = 32400.