The integer part of a real number $x$ is the greatest integer that is less than or equal to $x$. We denote the integer part of $x$ by $[x]$. For example, $[2.9]=2$, $[0.88]=0$, and $[-1.7]=-2$.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-025.jpg?height=128&width=808&top_left_y=604&top_left_x=704)

Calculate the following integer parts.
(a) $[\sqrt{12}]$
(b) $\left[\frac{28756}{12777}\right]$
(c) $\left[-\frac{2007}{2008}\right]$
(d) $[\sqrt[3]{-111}]$
(a) The numbers 9 and 16 are perfect squares and \(9 < 12 < 16\). Therefore,

\[
3 = \sqrt{9} < \sqrt{12} < \sqrt{16} = 4
\]

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-080.jpg?height=97&width=432&top_left_y=588&top_left_x=1326)

and, therefore, \([\sqrt{12}] = 3\).

(b) Since \(12777 \times 2 < 28756 < 12777 \times 3\), we have

\[
2 < \frac{28756}{12777} < 3, \text{ therefore, } \left[\frac{28756}{12777}\right] = 2
\]

(c) Since \(2007 < 2008\), we have \(0 < \frac{2007}{2008} < 1\), or

\[
-1 < -\frac{2007}{2008} < 0, \text{ therefore, } \left[-\frac{2007}{2008}\right] = -1
\]

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-080.jpg?height=388&width=432&top_left_y=882&top_left_x=1326)

(d) Since \(4^3 = 64 < 111 < 125 = 5^3\), we have

\[
(-5)^3 = -5^3 = -125 < -111 < -4^3 = (-4)^3
\]

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-080.jpg?height=94&width=432&top_left_y=1432&top_left_x=1326)

so that \(-5 < \sqrt[3]{-111} < -4\) and, therefore, \([\sqrt[3]{-111}] = -5\).