We need to find a number

k

such that its square root, cube root, and fourth root are positive whole numbers.

k = a 3 k = b 4 k = c

In the expressions above,

a, b,

and

c

are positive whole numbers. Using the definition of the

n^{th}

root of a number, we obtain the following relations.

k = a^{2} k = b^{3} k = c^{4}

Notice that

k

is a number that can be written as three different powers.

Additionally,

c

must be multiplied by itself

4

times to get the value of

k,

while

a

and

b

require fewer multiplications by themselves. This leads us to the following relation.

c < b < a

We can also set the following identities.

c^{4} = b^{3} = a^{2}

Since

c

is the smallest value, we will substitute some values for

c

and calculate

k .

After that, our mission will be to find the numbers

a

and

b .

c = 2 \Rightarrow c^{4} = 2^{4} = 16 = k

Next, we will try to write

16

as a whole number

b

cubed and as another whole number

a

squared.

Since $16$ cannot be written as a whole number cube, $c$ cannot be $2 .$ Let's make a table where we try some more numbers.

$c$	$c^{4} = k$	$a^{2} = k$	$b^{3} = k$
$3$	$3^{4} = 81$	$9^{2} = 81$	No way
$4$	$4^{4} = 256$	$1 6^{2} = 256$	No way
$5$	$5^{4} = 625$	$2 5^{2} = 625$	No way
$6$	$6^{4} = 1296$	$3 6^{2} = 1296$	No way
$7$	$7^{4} = 2041$	$4 9^{2} = 2041$	No way
$8$	$8^{4} = 4096$	$6 4^{2} = 4096$	$1 6^{3} = 4096$

From the table above, one number that satisfies the required condition is $4096 .$

Using Rational Exponents

We need to find a number such that its square, cube, and fourth roots are positive whole numbers.

k = whole number 3 k = whole number 4 k = whole number

Let's write the number we are looking for as

a^{b},

where

a

and

b

are positive whole numbers. Next, we will rewrite the roots as rational exponents.

a^{b / 2} = whole number a^{b / 3} = whole number a^{b / 4} = whole number

Here the key is the indexes of the roots, or equivalently, the denominators of the exponents:

2,

3,

and

4 .

Since

a

is a whole number, we need each exponent to be a whole number too.

\frac{b}{2} = whole number \frac{b}{3} = whole number \frac{b}{4} = whole number

From the above, we need

b

to be divisible by

2,

3,

and

4 .

One number that satisfies this condition is the least common multiple (LCM) of these numbers.

21231342122 \Rightarrow LCM (2, 3, 4) = 2 \cdot 3 \cdot 2 = 12

Consequently, any whole number

a

raised to

b = 12

will have a positive whole number for a square, cube, and fourth root. We can check it by trying some values.

$k$	$k$	$3 k$	$4 k$
$2^{12} = 4096$	$4096 = 64$	$34096 = 16$	$44096 = 8$
$3^{12} = 531441$	$531441 = 729$	$3531441 = 81$	$4531441 = 27$
$4^{12} = 16777216$	$16777216 = 4096$	$316777216 = 256$	$416777216 = 64$

Therefore, any number raised to the $1 2^{th}$ power will satisfy the given condition. Particularly, we can pick $4096 .$

Exercise