Key Points

A square number is the result of multiplying a number by itself, represented as $n^2 = n \times n$. The squares of natural numbers ($1, 4, 9, 16, \dots$) are called perfect squares.

A perfect square can only end with the digits $0, 1, 4, 5, 6,$ or $9$. If a number ends in $2, 3, 7,$ or $8$, it is never a perfect square.

A perfect square must have an even number of zeros at the end. For example, $100$ ($10^2$) is a perfect square, but $1000$ is not.

The sum of the first $n$ odd natural numbers is equal to $n^2$. For example, the sum of the first 4 odd numbers is $1 + 3 + 5 + 7 = 16$, which is $4^2$.

The square root is the inverse operation of squaring. If $n^2 = m$, then the square root of $m$ is $n$, written as $\sqrt{m} = n$. For example, since $8^2 = 64$, then $\sqrt{64} = 8$.

To find the square root of a perfect square, express it as a product of prime factors. The square root is the product of one factor from each pair. For example, $144 = 2^2 \times 2^2 \times 3^2$, so $\sqrt{144} = 2 \times 2 \times 3 = 12$.

A number is a perfect square if you can repeatedly subtract successive odd numbers ($1, 3, 5, \dots$) from it and reach 0. The number of subtractions performed gives the square root.

There are $2n$ non-perfect square numbers between the squares of two consecutive numbers, $n$ and $n+1$. That is, between $n^2$ and $(n+1)^2$, there are $2n$ numbers.

A cube number is the result of multiplying a number by itself three times, represented as $n^3 = n \times n \times n$. The cubes of natural numbers ($1, 8, 27, 64, \dots$) are called perfect cubes.

The cube root is the inverse operation of cubing. If $n^3 = m$, then the cube root of $m$ is $n$, written as $\sqrt[3]{m} = n$. For example, since $5^3 = 125$, then $\sqrt[3]{125} = 5$.

To find the cube root of a perfect cube, express it as a product of prime factors. The cube root is the product of one factor from each triplet. For example, $216 = 2^3 \times 3^3$, so $\sqrt[3]{216} = 2 \times 3 = 6$.

The square or cube of an even number is always even (e.g., $6^2 = 36, 4^3 = 64$). The square or cube of an odd number is always odd (e.g., $7^2 = 49, 5^3 = 125$).

To find the smallest number to multiply a given number by to make it a perfect square, find its prime factorization. Multiply by the factors that are not in pairs. For example, $72 = 2^3 \times 3^2 = (2 \times 2) \times 2 \times (3 \times 3)$. Multiply by 2 to get $144 = 12^2$.

To find the smallest number to multiply a given number by to make it a perfect cube, find its prime factorization. Multiply by the factors needed to complete each triplet. For example, $108 = 2^2 \times 3^3$. Multiply by 2 to get $216 = 6^3$.