# Perfect Squares, Cubes, and Powers

A **perfect square** is an integer that can be expressed as the product of two equal integers. For example, \(100\) is a perfect square because it is equal to \(10\times 10\). If \(N\) is an any integer, then \(N^2\) is a perfect square. Because of this definition, perfect squares are always non-negative.

Similarly, a **perfect cube** is an integer that can be expressed as the product of three equal integers. For example, \(27\) is a perfect cube because it is equal to \(3\times 3 \times 3.\) Determining if a number is a perfect square, cube, or higher power can be determined from the prime factorization of the number.

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## Perfect Squares

## Make a list of \(10\) perfect squares from the smallest.

We have \[\begin{array} &0^2=0, &(\pm1)^2=1, &(\pm2)^2=4, &(\pm3)^2=9, &(\pm4)^2=16, \\ (\pm5)^2=25, &(\pm6)^2=36, &(\pm7)^2=49, &(\pm8)^2=64, &(\pm9)^2=81 .\end{array}\] Thus, the answer is \[0, 1, 4, 9, 16, 25, 36, 49, 64, 81. \ _\square\]

## Find the differences between two adjacent perfect squares, 9 of them from the smallest. For example, start from \[\]

\[1^2-0^2=1-0=1.\]

We have \[\begin{array} &1^2-0^2=1-0=1, &2^2-1^2=4-1=3, \\3^2-2^2=9-4=5, &4^2-3^2=16-9=7,\\ 5^2-4^2=25-16=9, &6^2-5^2=36-25=11, \\ 7^2-6^2=49-36=13, &8^2-7^2=64-49=15, \\ 9^2-8^2=81-64=17 .\end{array}\] Thus, the answers are \[1, 3, 5, 7, 9, 11, 13, 15, 17. \ _\square\]

## Which of the following is NOT a perfect square? \[\]

\[\begin{array} &(a)~ 125 &&&(b)~ 144 &&&(c)~ 441 &&&(d)~ 225 \end{array}\]

Since \(144=12 \times 12, 441=21 \times 21\) and \(225=15 \times 15,\) none of \((b), (c)\) and \((d)\) is the answer. Since \(125=5 \times 5 \times 5=25 \times 5,\) which is not a perfect square but a perfect cube. So, The answer is \((a).\) \(_\square\)

## In the following equation, \(a, b\) and \(c\) are all distinct positive integers: \[\]

\[a^2+b^2=c^2.\]

## What is the smallest possible value of \(c?\)

Observe that \[3^2+4^2=5^2 \Rightarrow 9+16=25.\] Then the answer is \(c=5.\) \( _\square\)

## What is the positive number \(a\) in the following equation: \[\] \[5^2+12^2=a^2?\]

Observe that \[5^2+12^2=25+144=169=13^2.\] Then the answer is \(a=13.\) \( _\square\)

## What are the perfect squares between \(301\) and \(399?\)

Observe that \[\begin{array} &(\pm17)^2=289, &(\pm18)^2=324, &(\pm19)^2=361, &(\pm20)^2=400.\end{array}\] Then the answers are \(324\) and \(361.\) \(_\square\)

## Perfect Cubes

When you cube something you multiply it by itself three times. For example, \(5^3 = 5 \times 5 \times 5 = 125\). Cubing a positive number will result in a positive number, while cubing a negative number will result in a negative number.

Make a list of 10 perfect non-negative cubes starting from the smallest.

We have,

\[\begin {array} & 0^3 = 0, & 1^3 = 1, & 2^3 = 8, & 3^3 = 27, & 4^3 = 64, \\ 5^3 = 125, & 6^3 = 216, & 7^3 = 343, & 8^3 = 512, & 9^3 = 729. \end {array}\]

Thus the answer is

\[0, 1, 8, 27, 64, 125, 216, 343, 512, 729. \ _\square\]

## Perfect Powers

A perfect power is the more general form of squares and cubes. Specifically, it is any number that can be written as the product of some non-negative integer multiplied by itself at least twice. In other words, it is of the form \(n^m\) for some integers where \(n\ge 0\) and \(m > 1.\)

The set of perfect powers is the union of the sets of perfect squares, perfect cubes, perfect fourth powers, and so on. The perfect powers less than or equal to \(100\) are \[0,1,4,8,9,16,25,27,32,36,49,64,81,100.\]

**Cite as:**Perfect Squares, Cubes, and Powers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/perfect-squares/