Whole numbers (integers)

Evaluate the following combined operation respecting the order of operations:

$5 - [ 3 - ( -4 + 7 ) \cdot 2 ] + (-8) : 4$

Work out the following combined calculation, paying special attention to square brackets and powers with negative bases:

$12 - [(-2)^3 + 5 \cdot (-3)] : (-1) + (-4)^2$

Work out the numerical value of the following expression containing absolute values:

$|-15 + 8| - |3 - 9| + |-2|$

Determine the sign and work out the exact value of the following powers of integers:

• $(-2)^4$

• $(-3)^3$

• $-5^2$

• $(-1)^{15}$

An industrial freezer is at a temperature of $-18$ °C. If due to a power failure the temperature rises by $5$ °C and, after repairing it, the motors manage to lower it by $12$ °C, what is the final temperature of the freezer?

The balance of a bank account at the beginning of the month is $-150$ €. Over the course of the week, the following transactions take place: a cash deposit of $400$ €, an electricity bill payment of $85$ €, an outgoing bank transfer of $230$ € and a maintenance fee charge of $12$ €. Write down the combined calculation and determine the final balance of the account.

An underwater drone carries out an ocean exploration. It begins its dive by descending $45$ m from the surface. Then it descends a further $120$ m to investigate a trench. After taking samples, it ascends $85$ m and stops to photograph a coral. Finally, it descends $30$ m more to return to the sea floor. At what depth is it relative to sea level?

In a basketball tournament, the standings are decided by the point difference (difference between points scored and points conceded). In the first four matches, a team's results were: $75-82$, $90-85$, $68-77$ and $85-80$. Express each match's point difference as an integer and work out the team's total point difference at the end of the fourth round.

The city of Rome was traditionally founded in $753$ BC and the Western Roman Empire fell in $476$ AD. Given that in the Gregorian historical calendar there is no year $0$ (it goes directly from $1$ BC to $1$ AD), write an operation using integers to work out exactly how many years passed between both historical events.

A computer program modifies a counter $C$ (whose initial value is $C = 0$) using a loop that repeats $100$ times (iterations numbered from $1$ to $100$). On even iterations, the program adds $5$ to the counter; on odd iterations, it subtracts $8$. Write an expression using integers that models the process and work out the exact value of the counter $C$ at the end of the $100$ iterations.