Question

Find the sum of square root of 2.89 and 1.69.

Answer 1

Hint-: Square root of a number is a number when multiplied together produces the original number. Square root of a number $x$ is denoted as \[\sqrt x \]and when the result is multiplied by itself produces \[\sqrt x \times \sqrt x = x\], produces the original number.
Square root of a number can be calculated either by the estimation method or by the prime factorization method.

Complete step by step solution: -
We can see the number whose square root is to be found are in decimal form so convert these numbers into a fraction number. To convert a decimal number into fraction convert that number into \[\dfrac{a}{b}\]form: hence we write \[2.89 = \dfrac{a}{b} = \dfrac{{289}}{{100}}\]
Now we can find their square root of 289 and 100 individually and their ratio is the desired result.
We can write \[100 = {10^2}\], so the square root will be \[\sqrt {{{10}^2}} = {10^{2 \times \dfrac{1}{2}}} = 10\]
Now use prime factorization method to find square root of 289:
\[
   17\underline {\left| {289} \right.} \\
   17 \\
 \]
Hence we can write \[\left( {289} \right) = 17 \times 17\]
Now make pair of same numbers of factors
\[\left( {289} \right) = \underline {17 \times 17} = 17\]
So square root of 289 is: \[\sqrt {289} = 17\]
Hence the square root of 2.89 is calculated as:

\[
  2.89 = \dfrac{{289}}{{100}} \\
  \sqrt {289} = \sqrt {\dfrac{{289}}{{100}}} \\
   = \dfrac{{17}}{{10}} = 1.7 \\
 \]
Similarly follow the same steps to find square root of 1.69, convert it into fraction form \[\dfrac{a}{b}\]
\[1.69 = \dfrac{a}{b} = \dfrac{{169}}{{100}}\]
Where \[\sqrt {100} = 10\]
Now prime factorize 169 to find its square root:
\[
   13\underline {\left| {169} \right.} \\
   13 \\
 \]
We can write 169 as \[\left( {169} \right) = 13 \times 13\]
Now make the pair of same numbers of the factors
\[\left( {169} \right) = \underline {13 \times 13} = 19\]
Hence the square root of\[\sqrt {169} = 13\]
So the square root of \[1.69\] is calculated as:
\[1.69 = \dfrac{{169}}{{100}}\]
\[
  \sqrt {1.69} = \sqrt {\dfrac{{169}}{{100}}} \\
   = \dfrac{{13}}{{10}} = 1.3 \\
 \]
Now we have to find the sum of these two numbers which is equal to:
\[\sqrt {2.89} + \sqrt {1.69} = 1.7 + 1.3 = 3\]


Note: The cube root of a number can either be found by using the estimation method or by factorization method. But the best and easy method of finding cube roots is the factorization method as this has fewer calculations and saves time as well.