Let $x=(0.\overline{6})(0.\overline{60} )$. when $x$ is written out as a decimal, what is the sum of the first $15$ digits after the decimal point?

32

Step-by-step explanation:

We begin by writing both $0.\overline{6}$ and $0.\overline{60}$ as fractions.

Let $r=0.\overline{6}$. Then $10r=6.\overline{6}=6+r$, so $9r=6$ and $r=\frac{6}{9}=\frac{2}{3}$.

Let $s=0.\overline{60}$. Then $100s=60.\overline{60}=60+s$, so $99s=60$ and $s=\frac{60}{99}=\frac{20}{33}$.

Therefore, $x = r\cdot s = \frac{2}{3}\cdot \frac{20}{33} = \frac{40}{99}$.

We can convert $\frac{40}{99}$ to a decimal by long division:

$$

\begin{array}{*{8}{[email protected]{}} l}

&& 0 & . & 4 & 0 & 4 & 04 & \ldots \\

\cline{2-9}

99 & | & 40 & . & 0 & 0 & 0 & 00 & \ldots \\

&& 39 && 6 \\

\cline{3-5}

&&&& 4 & 0 & 0 \\

&&&& 3 & 9 & 6 \\

\cline{5-7}

&&&&&& 4 & 00 \\

&&&&&& 3 & 96 \\

\cline{7-8}

&&&&&&& 4 \\

&&&&&&& \vdots

\end{array}

$$

The repeating pattern of remainders explains why the decimal for $\frac{40}{99}$ will continue to repeat, giving us

$$\frac{40}{99} = 0.\overline{40} = 0.404040404040404\ldots~.$$The first $15$ digits after the decimal point are eight $4$s and seven $0$s, so their sum is $8\cdot 4 = \boxed{32}$.

answer:

it is general formula that is used to write value to its excet position

a11 a12 a13

a21 a22 a23

it is the example of order 2 by 3 matrix in this

a11 means value occur in 1 st row 1st column

a12 means value occurs in 1st row 2nd column

a13 means value occur in 1st row and 3rd column

a21 means value occurs in 2nd row and 1st column

a22 means value occurs in 2nd row and in 2nd column

a23 means value occurs in 2nd row and in 3rd column

slope= 1

y-inter slope= 0

x       y

o     0

1       1

2     2

ect.

step-by-step explanation: