Question

If √2=1.414 and√5=2.236 find the value of the (√10-√5 )÷2√2 upto three places of decimals.

Answer 1

Given that,
√2 = 1.414
√5 = 2.236

Rationalising the denominator means dividing something to the numerator and denominator so that denominator can be a rational number.

We've to rationalise
$\frac{ \sqrt{10} - \sqrt{5} }{2 \sqrt{2} }$

So, Let us divide √2 by the Numerator and the Denominator since we know that √2 multiplied by √2 gives us 2.

$= \frac{ \sqrt{10} - \sqrt{5} }{2 \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } \\ \\ = \frac{ \sqrt{2} ( \sqrt{10}) - \sqrt{2}( \sqrt{5} ) }{2 \sqrt{2} \times \sqrt{2} } \\ \\ = \frac{ \sqrt{20} - \sqrt{10} }{2 {( \sqrt{2} )}^{2} } \\ \\ \bf It \: can \: be \: also \: written \: as : \\ \\ = \frac{ \sqrt{2 \times 2 \times 5 } - \sqrt{2} \times \sqrt{5} }{2(2)} \\ \\ = \frac{2 \sqrt{5} - \sqrt{2} \times \sqrt{5} }{4} \\ \\ \bf Putting \: the \: given \: values \: we \: get \\ \\ = \frac{2(2.236) - (1.414)(2.236)}{4} \\ \\ = \frac{4.472 - 3.161}{4} \\ \\ = \frac{1.311}{4} \\ \\ = 0.327$

If any doubt, please ask ;)

Answer 2

$\underline \bold {Given:-}$

✓2 = 1.414

✓5 = 2.236

$\underline \bold {Solution:-}$

$\frac{ \sqrt{10} - \sqrt{5} }{2 \sqrt{2} } \\ \\ = \frac{ \sqrt{2 \times 5} - \sqrt{5} }{2 \sqrt{2} } \\ \\ = \frac{ \sqrt{2} \sqrt{5} - \sqrt{5} }{2 \sqrt{2} } \\ \\ = \frac{ \sqrt{5}( \sqrt{2} - 1) }{2 \sqrt{2} } \\ \\ on \: putting \: there \: values \\ \\ = \frac{2.236(1.414 - 1)}{2 \times 1.414} \\ \\ = \frac{1.118 \times 0.414}{1.414} \\ \\ = 0.327$