You need to find three positive whole numbers a, b, and c such that a^2=b^3=c^4.
Example Solution: 4096
Practice makes perfect
We need to find a number k such that its square root, cube root, and fourth root are positive whole numbers.
sqrt(k) = a sqrt(k) = b sqrt(k) = c
In the expressions above, a, b, and c are positive whole numbers. Using the definition of the n^\text{th} root of a number, we obtain the following relations.
k = a^2 k = b^3 k = c^4Notice 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 ⇒ 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
16^2=256
No way
5
5^4 = 625
25^2=625
No way
6
6^4 = 1296
36^2 = 1296
No way
7
7^4 = 2041
49^2 = 2041
No way
8
8^4 = 4096
64^2=4096
16^3=4096
From the table above, one number that satisfies the required condition is 4096.
Alternative Solution
Using Rational Exponents
We need to find a number such that its square, cube, and fourth roots are positive whole numbers.
sqrt(k) = whole number
sqrt(k) = whole number
sqrt(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.
b/2 = whole number [0.8em]
b/3 = whole number [0.8em]
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.
r|l
2 & 2
1
r|l
3 & 3
1
r|l
4 & 2
2 & 2
1
⇒ LCM( 2, 3, 4) = 2 * 3 * 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
sqrt(k)
sqrt(k)
sqrt(k)
2^(12) = 4096
sqrt(4096)=64
sqrt(4096) = 16
sqrt(4096) = 8
3^(12)=531 441
sqrt(531 441)=729
sqrt(531 441) = 81
sqrt(531 441) = 27
4^(12) = 16 777 216
sqrt(16 777 216) = 4096
sqrt(16 777 216) = 256
sqrt(16 777 216) = 64
Therefore, any number raised to the 12^(th) power will satisfy the given condition. Particularly, we can pick 4096.
