Find the first digit after the decimal point

[anemone]

Determine the first decimal digit after the decimal point in the number $\sqrt{x^2+x+1}$ if $\large x=2014^{2014^{2014}}$

 

[kaliprasad]

Determine the first decimal digit after the decimal point in the number $\sqrt{x^2+x+1}$ if $\large x=2014^{2014^{2014}}$

Spoiler

x is too large $2014^{2014^{2014}}$

so $x^2 + x + 1= (x + 1/2)^2 + 3/4$
= $(x+1/2)^2( 1+ 3/(4(x + 1/2)^2)$
so square root = $(x+1/2) ( 1 + 3/(8(x+1/2)^2) + ...)$
the term $3/(8(x+1/2)^2)$ is extremley small so << .1
so square root is x + 1/2 or 5 is the 1st digit after decimal

 

[anemone]

Spoiler

x is too large $2014^{2014^{2014}}$

so $x^2 + x + 1= (x + 1/2)^2 + 3/4$
= $(x+1/2)^2( 1+ 3/(4(x + 1/2)^2)$
so square root = $(x+1/2) ( 1 + 3/(8(x+1/2)^2) + ...)$
the term $3/(8(x+1/2)^2)$ is extremley small so << .1
so square root is x + 1/2 or 5 is the 1st digit after decimal

Hey kaliprasad, thanks for participating!:) Well done! Your answer is correct... but I think this edited version of the solution isn't quite straightforward than the before edited post.:p

 

[kaliprasad]

Hey kaliprasad, thanks for participating!:) Well done! Your answer is correct... but I think this edited version of the solution isn't quite straightforward than the before edited post.:p

this edited post is more accurate as it defines the reason. As you have made a reference to un edited post I mention the unedited post( exact words I do not remeber so in lines as below) which is highly informal

Spoiler

x is too large $2014^{2014^{2014}}$

so $x^2+x+1=(x+1/2)^2+3/4$ and as 3/4 is too small we can igmore so

square root =x + 1/2 so 1st digit after decimal = 5

 

[CellCoree]

To find the first decimal digit after the decimal point in the given expression, we can use the fact that the square root of a number is less than or equal to the number itself. Therefore, we can approximate the value of $\sqrt{x^2+x+1}$ by using the value of $x=2014^{2014^{2014}}$.

Using a calculator, we can find that $x$ is a very large number, approximately $2.96 \times 10^{2050}$. This means that the expression $\sqrt{x^2+x+1}$ will also be a very large number, but it will still have a decimal component.

To find the first decimal digit after the decimal point, we can use scientific notation. The value of $x$ can be rewritten as $2.96 \times 10^{2050}$. We can then use the properties of exponents to rewrite the expression as:

$\sqrt{x^2+x+1} = \sqrt{(2.96 \times 10^{2050})^2 + (2.96 \times 10^{2050}) + 1} = \sqrt{8.7616 \times 10^{4100} + 2.96 \times 10^{2050} + 1}$

Using a calculator, we can find that the value inside the square root is approximately $8.7616 \times 10^{4100}$. When we take the square root of this value, we get approximately $2.96 \times 10^{2050}$.

Therefore, the first decimal digit after the decimal point in the given expression is 9. This means that the value of $\sqrt{x^2+x+1}$ is approximately $2.96 \times 10^{2050}.9$.