Question

Evaluate $\sqrt {0.8} $ up to two places of decimal.

Answer 1

Hint: We will first express $\sqrt {0.8} $ as a variable. Then we will square on both sides to remove the square root. Next, we will express the resulting value as a rational number and multiply and divide the rational number by 10. Finally, we will express the numerator and denominator in the rational number as a square and take the square root.

Complete step-by-step answer:
Let us take $x = \sqrt {0.8} $.
 We shall square on both sides to get ${x^2} = 0.8$.
We can write $0.8{\text{ as }}\dfrac{8}{{10}}$ because there is only one digit after the decimal in 0.8.
 Thus, we have ${x^2} = \dfrac{8}{{10}}$
Now, we will multiply and divide by 10 on the LHS i.e., ${x^2} = \dfrac{8}{{10}} \times \dfrac{{10}}{{10}} = \dfrac{{80}}{{100}}$ ………………(1)
We will now prime factorise the numerator i.e., 80 and express it as a square. Thus,
$80 = 8 \times 10$
$ \Rightarrow 80 = 2 \times 2 \times 2 \times 2 \times 5$
Rewriting the multiples as exponent, we get
$ \Rightarrow 80 = {2^2} \times {2^2} \times 5 \\
   \Rightarrow 80 = {(2 \times 2 \times \sqrt 5 )^2} = {(4\sqrt 5 )^2} \\ $
Also, we know that the denominator i.e., $100 = {10^2}$. Substituting these values in equation (1), we get
${x^2} = \dfrac{{{{(4\sqrt 5 )}^2}}}{{{{(10)}^2}}}$ ………………………..(2)
 Now, we will take square root on both sides of equation (2). Hence,
$\Rightarrow$ $x = \dfrac{{4\sqrt 5 }}{{10}}$
We know that $\sqrt 5 = 2.236$.
Multiplying this value by 4 in the numerator, we get
$\Rightarrow$ $x = \dfrac{{4 \times 2.236}}{{10}} = \dfrac{{8.944}}{{10}}$.
 Dividing $8.944{\text{ by }}10$, we get $x = 0.8944$.
Since we have to evaluate $\sqrt {0.8} $ up to two decimal places, we finally get $\sqrt {0.8} = 0.89$.

Note: The above problem can also be approached by using the long division method. Since there is only one digit after the decimal place in 0.8, we fill the remaining places with 5 zeros and pair digits starting from right. Since the integral part in 0.8 is 0, the integral part in the square root will also be 0.