# Perfect Squares, Cubes, and Powers | Brilliant Math & Science Wiki (2023)

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 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.

#### Contents

• Perfect Squares
• Perfect Cubes
• Perfect Powers

## Perfect Squares

Make a list of $$10$$ perfect squares from the smallest.

We have

$\begin{array} &0^2=0, &(\pm1)^2=1, &(\pm2)^2=4, &(\pm3)^2=9, &(\pm4)^2=16, \\ (\pm5)^2=25, &(\pm6)^2=36, &(\pm7)^2=49, &(\pm8)^2=64, &(\pm9)^2=81 .\end{array}$

$0, 1, 4, 9, 16, 25, 36, 49, 64, 81. \ _\square$

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

$\begin{array} &1^2-0^2=1-0=1, &&&2^2-1^2=4-1=3, \\3^2-2^2=9-4=5, &&&4^2-3^2=16-9=7,\\ 5^2-4^2=25-16=9, &&&6^2-5^2=36-25=11, \\ 7^2-6^2=49-36=13, &&&8^2-7^2=64-49=15, \\ 9^2-8^2=81-64=17 .\end{array}$

$1, 3, 5, 7, 9, 11, 13, 15, 17. \ _\square$

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. Now, $$125=5 \times 5 \times 5=25 \times 5,$$ 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 (This is part of the Pythagorean Theorem):

$a^2+b^2=c^2.$

What is the smallest possible value of $$c?$$

Observe that

$3^2+4^2=5^2 \implies 9+16=25.$

Then the answer is $$c=5.$$ $$_\square$$

(Video) Perfect Squares Song- Aceon Academy

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 $$361.$$ $$_\square$$

Some properties regarding perfect squares are as follows (their proofs are omitted here):

1. Perfect squares cannot have a units digit of 2, 3, or 7. (You can check them out for yourself.)
2. The square of an even number is even and the square of an odd number is odd.
3. All odd squares are of the form $$4n + 1$$, hence all odd numbers of the form $$4n+3$$, where $$n$$ is a positive integer, are not perfect squares. For instance, 361 can be written as $$4 \times 90 + 1,$$ and we know $$361 = 19^2.$$ However, 843 is not a perfect square since $$29^{2}=841$$ and $$30^{2} = 900;$$ it can be expressed as $$4 \times 210 + 3.$$
4. All even numbers of the form $$4n + 2$$, where $$n$$ is a positive integer, are not perfect squares
5. All even squares are divisible by 4. (You can take any even square and check this.)
6. The difference of 2 odd squares is a multiple of 8. For example, $$15^{2} - 11^{2} =104,$$ which is $$8 \times 13.$$
7. The sum of the first $$n$$ odd numbers is in fact $$n^2.$$ For example, $$1+3+5+7+9+11= 36.$$ Here, there are 6 odd numbers, so we can find the sum as just $$n^2=6^2.$$ Similarly, $$1+3+5+\cdots+57 = 784,$$ as $$n = 28$$ here.
8. The sum of the first $$n$$ perfect squares $$1^{2} + 2^{2} + 3^{2} +\cdots+n^{2}$$ is given by $$\frac{n(n+1)(2n+1)}{6}.$$
9. If $$p$$ divides $$a^{2},$$ then $$p$$ divides $$a$$ as well (Euclid's theorem). From this, we can say that a number is a perfect square if its prime factorization contains all primes raised to some even power.
10. Given two positive integers $$K$$ and $$m,$$ if $$K^2-m$$ is the square of an integer $$n,$$ then $$K-n$$ divides $$m.$$

Ending digits for squared numbers (we consider decimal system):

• If a number has units digit 1 or 9, its square will have units digit 1.
• If a number has units digit 2 or 8, its square will have units digit 4.
• If a number has units digit 3 or 7, its square will have units digit 9.
• If a number has units digit 4 or 6, its square will have units digit 6.
• If a number has units digit 5, its square will have units digit 5.
• If a number has units digit 0, its square will have units digit 0.

The proof for this is left for the reader.

## 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

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

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

Some simple properties of perfect cubes are given below. The proofs for them are omitted here.

1. Every perfect cube has digital root 1, 8, or 9. By "digital root" we mean the sum of digits that is done until we get a single digit. For instance, the digital root of 1234 can be obtained as follows: We have $$1+2+3+4=10.$$ Since we got a 2-digit number, we add the digits again to get $$1+0= 1,$$ which is the digital root of 1234. The number 54 has digital root $$5+4 = 9.$$ Note that if a number has digital root 1, 8, or 9, it does not necessarily mean that it is a perfect cube (as is the case with 54, which has digital root 9 but is not a perfect cube). This can be proven with modular arithmetic.
2. Perfect cubes can have any number from 0 to 9 as their units digit.
3. The sum of the first $$n$$ perfect cubes $$1^{3} + 2^{3} + 3^{3} + 4^{3} +\cdots+ n^{3}$$ is $$\left(\frac{n(n+1)}{2}\right)^{2}.$$ This is equivalent to the square of the sum of the first $$n$$ natural numbers.
4. Every positive rational number can be expressed as the sum of three cubes of rational numbers.
5. It is possible to express any perfect cube as the sum of four odd numbers. For example, $$64= 13+15+17+19.$$

Units digits of perfect cubes:

• If a number ends in 0, its cube ends in 0.
• If a number ends in 2, its cube ends in 8.
• If a number ends in 3, its cube ends in 7.
• If a number ends in 4, its cube ends in 4.
• If a number ends in 5, its cube ends in 5.
• If a number ends in 6, its cube ends in 6.
• If a number ends in 7, its cube ends in 3.
• If a number ends in 8, its cube ends in 2.
• If a number ends in 9, its cube ends in 9.

The proof for this is left for the reader.

Only 6 of the following 7 numbers are perfect cubes. Which one is not?

$8000,\ 15625,\ 40323,\ 132651,\ 941192,\ 103823,\ 42875$

By applying the $$1^\text{st}$$ property, the digital root of 40323 is $$4+0+3+2+3= 12 \longrightarrow 1+2=3.$$ Thus, 40323 is not a perfect cube. $$_\square$$

(Video) Factoring Sums and Differences of Perfect Cubes

## 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 $$n\ge 0$$ 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

$0,1,4,8,9,16,25,27,32,36,49,64,81,100.$

A few simple results are given below. The proofs for these are omitted.

• The $$n^\text{th}$$ power of a number with units digit 5 will have units digit 5 again.
• The $$n^\text{th}$$ power of a number with units digit 1 will have units digit 1 again.
• The $$n^\text{th}$$ power of a number with units digit 0 will have units digit 0.
• The $$n^\text{th}$$ power of a number with units digit 6 will have units digit 6.
• The units digit of a number raised to 5 is the units digit of the original number. In fact, $$a^5=10{m} + a,$$ where $$a$$ and $$m$$ are positive numbers (which comes from a famous theorem by Euler).
• 0 to the $$n^\text{th}$$ power is 0, where $$n$$ is not equal to zero. 0 to the zeroth power is undefined.
• 1 to the $$n^\text{th}$$ power is 1.
• The number of digits in 10 to the power $$n$$ is $$n+1,$$ where there will be $$n$$ zeroes.
• Every $$4^\text{th}, 6^\text{th}, 8^\text{th}, ...., (2n)^\text{th}$$ power of a positive integer is a perfect square.

Which is the greatest of the following?

$2^8,\ 3^6,\ 7^4,\ 5^4,\ 10^3,\ 4^6?$

We have

$\begin{array} 2^8 = 512, & 3^6 = 729, & 7^4 = 2401, & 5^4 = 625, & 10^3 = 1000, &4^6 = 4096. \end{array}$

Thus $$4^6$$ is the greatest among these numbers. Do note that there are better ways to determine which is the greatest or least given a set of numbers; the above powers can all be easily computed.

Find the number of digits in $$64^{7}$$, given $$6^{7} = 279936$$ and $$7^{7}= 823543.$$

Since we are given $$6^{7}$$ and $$7^{7}$$, we can easily see that the number of digits of any positive integer between 60 and 70 is in fact 13. Here's how:If $$n$$ is greater than $$m,$$ then $$n$$ raised to $$k$$ is greater than $$m$$ raised to $$k,$$ where $$n$$ and $$m$$ are real numbers and $$k$$ is a positive integer. Knowing this, we can find the number of digits in $$64^{7}:$$

$\begin{array} 60^7 = 6^7 \times 10000000 = 2799360000000, & 70^7 = 7^7 \times 10000000 = 8235430000000. \end{array}$

Since both $$60^{7}$$ and $$70^{7}$$ have an equal number of digits and $$60^{7}$$ is less than $$64^{7}$$ but $$64^{7}$$ is less than $$70^{7}$$, $$64^{7}$$ must also have an equal number of digits. Hence $$64^{7}$$ has 13 digits. $$_\square$$

(Video) How to Learn Your Cubic Numbers

• Square Roots
• Number Theory
• Finding the Last Digit of a Power
• Cube Roots
(Video) 2000 years unsolved: Why is doubling cubes and squaring circles impossible?

## FAQs

### What are perfect squares and perfect cubes? ›

Perfect squares are the squares of a whole number (when a number is multiplied by itself two times). Perfect cubes are the cubes of a whole number (when a number is multiplied by itself three times).

What are cubes and perfect cubes? ›

Therefore, if we take the cube root of a perfect cube, we get a natural number and not a fraction. Hence, 3√x = y. For example, 8 is a perfect cube because 3√8 = 2.
...
List of Perfect Cubes 1 to 50.
Number (x)Multiplied Three times by itselfCubes (x3)
88× 8× 8512
99× 9× 9729
1010× 10× 101000
1111× 11× 111331
46 more rows
Aug 25, 2020

How many perfect squares and cubes are there? ›

Explanation − 1, 64 and 4096 are only numbers from 1 to 5000 that are both perfect squares and cubes.

What is perfect squares in math? ›

Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.

What are 3 examples of perfect squares? ›

They are 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961.

What is perfect square theory? ›

A perfect square is an integer that can be expressed as the product of two equal integers. For example, 100 100 is a perfect square because it is equal to 10 × 10 10\times 10 10×10. If N is an integer, then N 2 N^2 N2 is a perfect square.

How do you solve perfect cubes? ›

So, yes, this is the difference of perfect cubes. Use formula: a3 - b3 = (a - b)(a2 + ab + b2). Be sure to use parentheses to avoid problems. Remove the inner parentheses on the trinomial expression.

How can we identify perfect cubes? ›

In order to check whether a number is a perfect cube or not, we find its prime factors and group together triplets of the prime factors. If no factor is left out then the number is a perfect cube. However if one of the prime factors is a single factor or a double factor then the number is not a perfect cube.

Is cubes a good math strategy? ›

C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work. Why I like it: Gives students a very specific 'what to do. '

What is the difference between squares and cubes? ›

The major difference between the square and the cube is the square is a two-dimensional figure and it has only two dimensions such as length and breadth, whereas the cube is a three-dimensional figure and its three dimensions are length, breadth and height. The cube is obtained from the shape square.

### What does a perfect cube mean? ›

The perfect cube is the result of multiplying the same integer three times. Multiplying the number 4 three times, for example, yields 64. As a result, the number 64 is a perfect cube. Perfect cube = number x number x number.

Is 64 a perfect square or cube? ›

Since 64 is a perfect cube, therefore it is easy to find its cube root, but for imperfect cubes we have to estimate the values.

Why are perfect squares important? ›

Perfect squares are important because they're an example of how to take the square root of a perfectly precise natural number. The square root of a perfect square must also be a natural number, meaning that it's a non-decimal, non-fractional integer.

How do you teach perfect squares? ›

A perfect square is the square of a whole number. Recall that squaring a number means to multiply it by itself. The perfect squares can be found along the diagonal of the multiplication table when a whole number is multiplied by itself. For example, the number 25 is a perfect square because 5 × 5 = 25 .

Why is it called a perfect square? ›

A perfect square is a number that can be expressed as the product of two equal integers. What does that mean? Basically, a perfect square is what you get when you multiply two equal integers by each other. 25 is a perfect square because you're multiplying two equal integers (5 and 5) by each other.

What is the difference between square and perfect square? ›

Hint: A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.
...
What is the difference between perfect square and difference of squares?
Perfect squareDifference of squares
Perfect square values cannot have 2, 3, 7 or 8 at the unit's place.Differences of squares may have 2, 3, 7 or 8 at the units place.
4 more rows
Mar 1, 2023

How do you know whether a number is a perfect square? ›

A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. A number that is a perfect square never ends in 2, 3, 7 or 8.

Where are perfect squares used in real life? ›

A chessboard is one of the best examples of a square-shaped object in daily life.

What is the cube rule in math? ›

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3.

What is the easiest cube to solve? ›

6 Best Rubik's Cube for Beginners
• Drift 3x3.
• Drift 4x4.
• Drift 3x3 Gear Cube.
• Drift Mirror Cube Blue.
• Drift Maple Leaf.
• Drift Ghost Cube.
Feb 2, 2021

### What is the perfect square formula? ›

When a polynomial is multiplied by itself, then it is a perfect square. Example – polynomial ax2 + bx + c is a perfect square if b2 = 4ac.

What is an example of a perfect cube in math? ›

Perfect cube numbers can be obtained by multiplying every number thrice by itself. For example, 1 × 1 × 1 = 1 and 2 × 2 × 2 = 8 and so on. The list of perfect cubes from 1 to 10 is as follows: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.

Can a negative number be a perfect cube? ›

Can a negative number be a perfect cube? Yes, a cube of any negative number is always a negative number.

What are the first 20 perfect cubes? ›

Cube Numbers from 1 to 100
NumberCube
196859
208000
219261
2210648
96 more rows

What is the smartest math problem? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

What is the best math strategy? ›

'Concrete Representational Abstract'

The single most effective strategy that I have used to teach mathematics is the Concrete Representational Abstract (CRA) approach. During the concrete step, students use physical materials (real-life objects or models) to explore a concept.

How do you know if a number is a square or a cube? ›

A square number is the result of multiplying a number by itself. For example, 5 × 5 (or 52) = 25 Therefore, 25 is a square number. A cube number is the result of multiplying a number by itself once, then twice. For example, 2 × 2 × 2 (or 23) = 8 Therefore, 8 is a cube number.

How many types of squares are there? ›

There are 3 types of squares, a different number of each type, and each with a different number of squares they share exactly one corner with: Corners - 4 of them-shares exactly 1 corner with 1 square.

Why is 144 not a perfect cube? ›

Is 144 a Perfect Cube? The number 144 on prime factorization gives 2 × 2 × 2 × 2 × 3 × 3. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 144 is irrational, hence 144 is not a perfect cube.

Why is 27 a perfect cube? ›

Is 27 a perfect cube number? Yes, 27 is a perfect cube number since the cube root value of 27 is a whole number.

### Why is the 128 perfect cube? ›

Doing Prime factorization of 128 We see that 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 Since 2 does not occur in triplets, ∴ 128 is not a perfect cube.

Is 729 a perfect square? ›

Hence, 729 is a perfect square number, and 729 is a square of 27.

Is 729 a perfect cube? ›

Is 729 a perfect cube? 3√729 = 9 and 9 is an integer. So, 729 is a perfect cube.

Is 289 a perfect square? ›

289 is a perfect square as the answer obtained after square root is a rational number. The square root of 289 can be simplified to 17 either by using the prime factorization of 289 or by expressing 289 as a square of 17.

Who invented perfect squares? ›

The Babylonians and Greeks have been credited with the discovery of Heron's method, the precursor of Newton's iterative method, although Indian mathematicians are thought to have used a similar system around 800BC.

What grade do you learn perfect squares? ›

Another concept you will learn in sixth grade math is perfect squares. Perfect squares are the squares of whole numbers (1, 4, 9, 16, 25, etc.).

Why are perfect squares irrational? ›

They cannot be written as ratios or fractions. They are decimals which never end or repeat. The square roots of perfect squares are rational numbers and can be place on a number line. The square roots of non-perfect squares are irrational numbers.

What are perfect squares in math for kids? ›

Perfect squares are numbers that are the products of integers by themselves. In other words, when an integer is multiplied by itself, the resulting product is termed as a perfect square of the given number. For example, 36 is a perfect square because it is the product of 6 by itself, 6×6 = 36.

Why is 8 not a perfect square? ›

All perfect squares end in 1, 4, 5, 6, 9 or 00 (i.e. Even number of zeros). Therefore, a number that ends in 2, 3, 7 or 8 is not a perfect square.

Why is 16 called a square number? ›

The result of multiplying an integer (not a fraction) by itself. Example: 4 × 4 = 16, so 16 is a square number.

### Are perfect squares always even? ›

Not all even numbers are perfect squares. In fact, many perfect squares are odd numbers. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. These perfect squares result from the numbers 1 to 10 being multiplied by themselves.

Is 27 a perfect square or cube? ›

Yes, 27 is a perfect cube number since the cube root value of 27 is a whole number.

Is 64 a perfect square or perfect cube? ›

Since 64 is a perfect cube, therefore it is easy to find its cube root, but for imperfect cubes we have to estimate the values.

Is 49 a perfect square or cube? ›

√49 = 7 - 7 is an integer → number 49 is a perfect square.

What is the highest perfect cube? ›

Perfect cube numbers can be obtained by multiplying every number thrice by itself. For example, 1 × 1 × 1 = 1 and 2 × 2 × 2 = 8 and so on. The list of perfect cubes from 1 to 10 is as follows: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.

How do you tell if a number is a perfect cube? ›

In order to check whether a number is a perfect cube or not, we find its prime factors and group together triplets of the prime factors. If no factor is left out then the number is a perfect cube. However if one of the prime factors is a single factor or a double factor then the number is not a perfect cube.

Why 784 is a perfect square? ›

A perfect square is a number that can be expressed as a square of any integer, 784 is a perfect square number as (± 28)2 = 784.

Is 512 a perfect cube? ›

Yes, 512 is a perfect cube number since the cube root value of 512 is a whole number.

Is 1331 a perfect cube? ›

A perfect cube is a number which is the cube of an integer. Since the cube of 11 is 1331, therefore 1331 is a perfect cube number.

Is 4096 a perfect cube or square? ›

Yes, 4096 is the perfect cube since its cube root is a whole number, i.e. 16.

### Is 111 a perfect square? ›

Is the number 111 a Perfect Square? The prime factorization of 111 = 31 × 371. Here, the prime factor 3 is not in the pair. Therefore, 111 is not a perfect square.

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