Question

If the square root of \[23\dfrac{{394}}{{729}} = \dfrac{a}{{27}}\] then find the value of a?

Answer 1

Hint: Here firstly we will convert the mixed fraction into improper fraction and then use the long division method to find the square root . Then we will compare it with the given quantity to get the value of a.

Long division method is used to divide a large number (usually three digits or more) by a number having two or more digits.
Steps to find square root by long division method:
1. Place a bar over the pair of numbers starting from the unit place or Right-hand side of the number.
2. Take the largest number as the divisor whose square is less than or equal to the number on the extreme left of the number. The digit on the extreme left is the dividend. Divide and write the quotient.
3. Next, we then bring down the number, which is under the bar, to the right side of the remainder
4. Now double the value of the quotient and enter it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend.
5. Continue the process till the remainder is zero and then write the quotient as the answer.

Complete step by step solution:
Considering Left hand side we get,
\[LHS = \sqrt {23\dfrac{{394}}{{729}}} \]
We will change this mixed fraction into improper fraction:
\[
  LHS = \sqrt {\dfrac{{\left( {23 \times 729} \right) + 394}}{{729}}} \\
  LHS = \sqrt {\dfrac{{16767 + 394}}{{729}}} \\
  LHS = \sqrt {\dfrac{{17161}}{{729}}} \\
 \]
Now we have to find the square root of the quantity in LHS.

Therefore calculating the square root of numerator using long division method we get:
\[
  {\text{ }}131 \\
   \cdots \cdots \cdots \cdots \cdots \\
  1\mathop{\left){\vphantom{1{17161}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{17161}}}}
\limits^{\displaystyle \,\,\, 1} \\
   - {\text{1}} \\
   \cdots \cdots \cdots \cdots \\
  23\mathop{\left){\vphantom{1{07161}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{07161}}}}
\limits^{\displaystyle \,\,\, 3} \\
  {\text{ }} - 69 \\
   \cdots \cdots \cdots \cdots \cdots \\
  261\mathop{\left){\vphantom{1{261}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{261}}}}
\limits^{\displaystyle \,\,\, 1} \\
  {\text{ }} - 261 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \cdots \\
 \]
Hence the square root of 17161 is 131. (1)
Now calculating the square root of denominator using long division method we get:
\[
  {\text{ 27}} \\
   \cdots \cdots \cdots \cdots \cdots \\
  2\mathop{\left){\vphantom{1{729}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{729}}}}
\limits^{\displaystyle \,\,\, 2} \\
   - 4 \\
   \cdots \cdots \cdots \cdots \\
  47\mathop{\left){\vphantom{1{329}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{329}}}}
\limits^{\displaystyle \,\,\, 7} \\
  {\text{ }} - 329 \\
   \cdots \cdots \cdots \cdots \cdots \\
  {\text{ 0}} \\
   \cdots \cdots \cdots \cdots \cdots \\
 \]
Hence the square root of 729 is 27.
Therefore we get the value of LHS as:
\[LHS = \dfrac{{131}}{{27}}\]
Now equating LHS with RHS we get:
\[
  \dfrac{{131}}{{27}} = \dfrac{a}{{27}} \\
  a = 131 \\
 \]

Hence the value of a is 131.

Note:
This question can also be done by prime factorization method.
We have to find the prime factors of each of the numerator and the denominator and then take the square root of both the quantities .Then equate with RHS to get the value of a.