Question

-√140 -14.76(repeating) 12.7

Answer 1

To simplify or operate on the provided expression \(-\sqrt{140} - 14.76\overline{76} + 12.7\), we can break it down step by step.

1. **Calculate \(-\sqrt{140}\)**:
\[
\sqrt{140} = \sqrt{4 \times 35} = 2 \sqrt{35}
\]
The approximate decimal value of \(\sqrt{140} \approx 11.83\) (you can use a calculator for precision).
Thus,
\[
-\sqrt{140} \approx -11.83
\]

2. **Interpret \( -14.76\overline{76} \)**:
The repeating decimal \(14.76\overline{76}\) can be expressed as:
\[
14.76767676\ldots
\]
To convert \(0.76\overline{76}\) into a fraction:
Let \(x = 0.76767676\ldots\).
Then, multiplying by 100, we have:
\[
100x = 76.76767676\ldots
\]
Subtracting the first equation from the second,
\[
100x - x = 76.76767676\ldots - 0.76767676\ldots
\]
This simplifies to:
\[
99x = 76 \implies x = \frac{76}{99} \approx 0.76767676
\]
Thus, \(14.76\overline{76} = 14.76 + 0.76767676 \approx 14.76 + 0.77 \approx 15.53.\)
Therefore,
\[
-14.76\overline{76} \approx -15.53.
\]

3. **Calculate \(12.7\)**.

Now, we add all of these terms together:
\[
-\sqrt{140} - 14.76\overline{76} + 12.7 \approx -11.83 - 15.53 + 12.7.
\]

Combining these:
\[
-11.83 - 15.53 + 12.7 \approx -15.66.
\]

Thus, the approximate value of the expression is:
\[
\boxed{-15.66}.
\]