I have holes in my math because I didn't pay attention when I was a kid.( so please explain in detail if possible <3 )
While relearning everything I found my self stuck not understanding how this works :
$$\sqrt{\frac{49}{81}}$$
my steps to solve this were :
1- divide 49 by 81
2- square root the result $$\sqrt{0.6049382716049383}$$
3-the result is 0.7777777777777778
so what i need to understand is :
1- isnt 0.777777777777778 = 0.8 ?
2- is 0.7777777777777778 ( my result ) = $\frac{7}{9}$ ( which is the correct result according to the book )
The short answer is that for question 1, the answer is no, absolutely not. For question 2, technically also no, but it's so close that for most practical purposes, yes.
In more detail, you have to think about what these decimals actually mean. Apologies for the lack of repeating notation, I'm writing on my phone.
0.77777777777778 is the result you get from a calculator, but that's only because your calculator can only do so much decimal precision. The actual answer is 0.7777... repeating infinite sevens forever. This is equal to 7/9.
But what you've done with the decimals is far too much work and over-relies on the calculator (imagine how hard this would be to do by hand). The problem can be done mentally. Just take the square root of the numerator and denominator. $$\sqrt{49/81}=\sqrt{49}/\sqrt{81}=7/9. $$
I'd also like to address why 0.7... repeating doesn't equal 0.8. That would be 0.8=0.799999999... with infinite repeating 9s. If you subtract these numbers, you'd get infinite repeating zeros, that is, 0. For two numbers to subtract to get 0, they must be equal. However, if you subtract 0.8 - 0.77777777... you'd get 0.0222222222... repeating. Since their difference isn't 0, they're not the same.
Hope this helps!