Question

How do you simplify \[\sqrt {22.5} \]?

Answer 1

Hint: We solve for the square root of the given number using the division method. Write the digits after the decimal in pairs of two each starting from the right side and keep making pairs till all the digits are covered in the number ( digits before the decimal too). Then finding the suitable multiple of the term we divide the given number.

Complete step-by-step solution:
We have to find the square root of the number 22.5
We can write the number 22.5 as 22.50 so we can create a pair on right side of the decimal as well.
First we pair the digits in the number after the decimal point in pair of two each starting from the right side. Similarly we pair the digits before the decimal place.
\[22.50 = \overline {22} .\overline {50} \]
Then we take the highest number whose square will be less than or equal to the first pair i.e. 22
So, we know, \[1 \times 1 = 1,2 \times 2 = 4,3 \times 3 = 9,4 \times 4 = 16,5 \times 5 = 25\]
We choose \[4 \times 4 = 16\] because \[16 < 22\]
Now we divide the number by taking this number as divisor and taking the same number as quotient.
\[\begin{gathered}
  4\mathop{\left){\vphantom{1{\overline {22} .\overline {50} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {22} .\overline {50} }}}}
\limits^{\displaystyle \,\,\, 4} \\
   - 16 \downarrow \\
  \overline { = 650} \\
\end{gathered} \]
We bring down the next pair of numbers after the decimal place and make it the new dividend.
Now the remainder becomes the next dividend and the new divisor is twice the old divisor followed by a digit which makes a number such that square of that number will be less than or equal to the new dividend. We place the decimal in the quotient and then write the next digit in the quotient.
So, we have new dividend as 650 and we can have divisor as \[2 \times 4\underline {} = 8\underline {} \]where blank is filled by the same digit.
Now we try to find a number in the lane of forties whose square is less than or equal to our new dividend.
\[\begin{gathered}
  81 \times 1 = 81 \\
  82 \times 2 = 164 \\
  83 \times 3 = 249 \\
  84 \times 4 = 336 \\
  85 \times 5 = 425 \\
  86 \times 6 = 516 \\
  87 \times 7 = 609 \\
\end{gathered} \]
We can clearly see that \[87 \times 7 = 609\] suits our requirement because \[609 < 650\]
Now we divide the dividend by the number 87 and the quotient 6comes after the decimal beside the earlier quotient.
\[\begin{gathered}
  87\mathop{\left){\vphantom{1{650}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{650}}}}
\limits^{\displaystyle \,\,\, {4.7}} \\
   - 609 \downarrow \\
  \overline { = 41} \\
\end{gathered} \]

\[\therefore \] The value of \[\sqrt {22.5} \] is 4.7

Note: Students might make mistakes while calculating the square root as they bring down all the elements after the decimal in the second step, keep in mind we made pairs of elements so we can bring each pair at a time.
Also, when writing the division we need not write decimal in new dividends as it will just cause confusion.