Natural numbers
You have been using natural numbers since 1st ESO. This section is a quick but thorough review of what you already know —order of operations, division, powers, roots— and adds a new tool: counting combinations without having to list them one by one.
Combined operations: a refresher on the order of operations
The order does not change: first brackets (parentheses, square brackets and curly brackets); then powers and roots; next multiplications and divisions; finally additions and subtractions. Within the same level, work from left to right.
Work out the result of the following combined operation with natural numbers, respecting the order of operations: $15 - 3 \cdot (4 + 2) + 2^3 \cdot 5$
Work out the result of the following combined operation with natural numbers, respecting the order of operations:
$15 - 3 \cdot (4 + 2) + 2^3 \cdot 5$
Integer division
Every integer division satisfies a fixed relationship between its four elements.
In an integer division of natural numbers, the divisor is $14$, the quotient is $23$ and the remainder is $5$. Work out the value of the dividend.
In an integer division of natural numbers, the divisor is $14$, the quotient is $23$ and the remainder is $5$. Work out the value of the dividend.
Halley's Comet orbits the Sun and is visible from Earth approximately every $76$ years. Its last recorded appearance was in the year $1986$. Will it be visible in the…
Halley's Comet orbits the Sun and is visible from Earth approximately every $76$ years. Its last recorded appearance was in the year $1986$. Will it be visible in the year $3050$? If not, what will be the closest year to $3050$ in which it can be observed?
Powers of natural numbers
Review the rules: they are used constantly to simplify large expressions without reaching for the calculator.
Simplify the following expression using index laws and express the result as a single power: $2^4 \cdot 5^4 \cdot 10^2$
Simplify the following expression using index laws and express the result as a single power:
$2^4 \cdot 5^4 \cdot 10^2$
Use prime factorisation and index laws to fully simplify the following expression, giving the result as a natural number: $\frac{18^3 \cdot 12^4}{24^2 \cdot 9^4}$
Use prime factorisation and index laws to fully simplify the following expression, giving the result as a natural number:
$\frac{18^3 \cdot 12^4}{24^2 \cdot 9^4}$
Observe the pattern of the powers of $7$ ($7^1=7$, $7^2=49$, $7^3=343$, $7^4=2401$, $7^5=16807$, ...). Use the pattern followed by the units digits to determine what the…
Observe the pattern of the powers of $7$ ($7^1=7$, $7^2=49$, $7^3=343$, $7^4=2401$, $7^5=16807$, ...). Use the pattern followed by the units digits to determine what the last digit of the number $7^{2025}$ will be.
Exact roots and perfect squares
A very useful trick: the difference between two consecutive squares is always an odd number.
Work out the exact value of the following operation with roots of natural numbers: $\sqrt{144} + \sqrt[3]{125} - \sqrt[4]{16}$
Work out the exact value of the following operation with roots of natural numbers:
$\sqrt{144} + \sqrt[3]{125} - \sqrt[4]{16}$
A farmer wants to plant $1000$ apple trees forming a perfect grid (a square with the same number of trees in each row and in each column). How many trees will make up…
A farmer wants to plant $1000$ apple trees forming a perfect grid (a square with the same number of trees in each row and in each column). How many trees will make up the side of the largest possible square? How many apple trees will be left unplanted?
A square mosaic is formed by square tiles of $1 \text{ cm} \times 1 \text{ cm}$. The artist decides to enlarge the mosaic by adding one more row and one more column of…
A square mosaic is formed by square tiles of $1 \text{ cm} \times 1 \text{ cm}$. The artist decides to enlarge the mosaic by adding one more row and one more column of tiles to the edges to form a larger square. If $41$ new tiles were needed for this extension, how many tiles made up the original mosaic?
Counting without listing: the multiplication principle
If a process has several independent stages, the total number of possible outcomes is the product of the number of options at each stage.
A restaurant offers a set menu consisting of $3$ first courses, $4$ main courses and $2$ different desserts. How many different menu combinations (first course, main…
A restaurant offers a set menu consisting of $3$ first courses, $4$ main courses and $2$ different desserts. How many different menu combinations (first course, main course and dessert) can be chosen?