Power Play (Exponents) | FIO

Question 13

64 is a square number ( $8^2$ ) and a cube number ( $4^3$ ). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Solution

Understand the Question

A perfect square is a number of the form $a^2$ , where the exponents of its prime factors are multiples of $2$ .
A perfect cube is a number of the form $b^3$ , where the exponents of its prime factors are multiples of $3$ .
For a number to be both a square and a cube, the exponents in its prime factorization must be multiples of both $2$ and $3$ , which means they must be multiples of $\text{LCM}(2, 3) = 6$ .
Thus, all such numbers can be described generally as perfect sixth powers of the form $k^6$ (where $k$ is a whole number).

Step 1 · Define Square and Cube Conditions

Let $N$ be a number that is both a perfect square and a perfect cube. Diagram 1

As a square, $N$ can be written as: $N = a^2$

As a cube, $N$ can be written as: $N = b^3$

where $a$ and $b$ are whole numbers.

Step 2 · Derive the General Form

Consider the prime factorisation of $N$ :

For $N$ to be a square, prime factor exponents must be even (e.g., $36 = 2^2 \times 3^2$ ).
For $N$ to be a cube, prime factor exponents must be multiples of $3$ (e.g., $216 = 2^3 \times 3^3$ ).

For $N$ to be both, the exponents must be common multiples of $2$ and $3$ : $\text{LCM}(2, 3) = 6$

Thus, $N$ must be of the form $k^6$ for some whole number $k$ .

Verifying $k^6$ as both a square and a cube: $k^6 = (k^3)^2 \implies \text{a perfect square where } a = k^3$ $k^6 = (k^2)^3 \implies \text{a perfect cube where } b = k^2$

Step 3 · List Examples of Such Numbers

Evaluating $N = k^6$ for whole numbers $k$ :

For $k = 1$ :

\begin{aligned} N &= 1^6 = 1 \\[0.6em] 1 &= 1^2 = 1^3 \end{aligned}

For $k = 2$ :

\begin{aligned} N &= 2^6 = 64 \\[0.6em] 64 &= 8^2 = 4^3 \end{aligned}

For $k = 3$ :

\begin{aligned} N &= 3^6 = 729 \\[0.6em] 729 &= 27^2 = 9^3 \end{aligned}

For $k = 4$ :

\begin{aligned} N &= 4^6 = (2^2)^6 = 2^{12} = 4096 \\[0.6em] 4096 &= 64^2 = 16^3 \end{aligned}

For $k = 5$ :

\begin{aligned} N &= 5^6 = 15625 \\[0.6em] 15625 &= 125^2 = 25^3 \end{aligned}

Answer

Yes, there are infinitely many such numbers. In general, they are perfect sixth powers of the form $k^6$ (where $k$ is a whole number), giving $1, 64, 729, 4096, 15625, \dots$

Common Mistakes

Adding Exponents Instead of Finding LCM: Assuming the power must be $2 + 3 = 5$ ( $k^5$ ) instead of the least common multiple $\text{LCM}(2, 3) = 6$ ( $k^6$ ).
Overlooking Smallest Cases: Forgetting that $0$ ( $0^6 = 0^2 = 0^3$ ) and $1$ ( $1^6 = 1^2 = 1^3$ ) are also valid numbers that are both squares and cubes.

More questions in FIO

Express the following in exponential form:

(i) $6 \times 6 \times 6 \times 6$

(ii) $y \times y$

(iii) $b \times b \times b \times b$

(iv) $5 \times 5 \times 7 \times 7 \times 7$

(v) $2 \times 2 \times a \times a$

(vi) $a \times a \times a \times c \times c \times c \times c \times d$

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Find out the units digit in the value of $2^{224} \div 4^{32}$ ? [Hint: $4 = 2^2$ ]

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?

Write the given number as the product of two or more powers in three different ways. The powers can be any integers.

(i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Examine each statement below and find out if it is 'Always True', 'Only Sometimes True', or 'Never True'. Explain your reasoning.

(i) Cube numbers are also square numbers.

(ii) Fourth powers are also square numbers.

(iii) The fifth power of a number is divisible by the cube of that number.

(iv) The product of two cube numbers is a cube number.

(v) $q^{46}$ is both a 4th power and a 6th power ( $q$ is a prime number).

Simplify and write these in the exponential form.

(i) $10^{-2} \times 10^{-5}$

(ii) $5^7 \div 5^4$

(iii) $9^{-7} \div 9^4$

(iv) $(13^{-2})^{-3}$

(v) $m^5 n^{12} (mn)^9$

If $12^2 = 144$ , what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Q10

Circle the numbers that are the same—

$2^4 \times 3^6 \qquad 6^4 \times 3^2 \qquad 6^{10} \qquad 18^2 \times 6^2 \qquad 6^{24}$

Q11

Identify the greater number in each of the following—

(i) $4^3$ or $3^4$

(ii) $2^8$ or $8^2$

(iii) $100^2$ or $2^{100}$

Q12

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits $0$ – $9$ , how many digits should the code consist of?

Q13

64 is a square number ( $8^2$ ) and a cube number ( $4^3$ ). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Q14

A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?

Q15

The worldwide population of sheep (2024) is about $10^9$ , and that of goats is also about the same. What is the total population of sheep and goats?

(i) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

(vi) $10^9 + 10^9$

Q16

Calculate and write the answer in scientific notation:

(i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing.

(ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

(iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world.

(iv) Total time spent eating in a lifetime in seconds.

Q17

What was the date 1 arab/1 billion seconds ago?