if the product 4795795472k represents a perfect fourth power, then the smallest possible natural value for k is
4795795472 = 2^4 * 13^2 * 17^3 * 19^2
so k must be 13^2 * 17 * 19^2 = _____
4795795472 = 2^4 x 13^2 x 17^3 x 19^2
Therefore, the smallest value for k has to be 13^2 x 17^1 x 19^2 = 1037153
Well, if the product 4795795472k represents a perfect fourth power, then k must be a natural number that, when multiplied by 4795795472, gives us a perfect fourth power.
Now, there are infinite possibilities for k, but let's try to find the smallest one.
To find a perfect fourth power, we need to check if there is a natural number n such that n^4 = 4795795472k.
Now, if we simplify this equation, we get n^4 = (2^12)(3^2)(7^2)(13^2)k.
To find the smallest possible value for k, we need to find the smallest n that satisfies this equation.
However, since there are several prime factors on the right side of the equation, it is unlikely that a small value of n can be found. Therefore, without performing further calculations, I can't determine the smallest possible natural value for k.
In order for the product 4795795472k to represent a perfect fourth power, we need to find the smallest possible value for k.
To do this, we can try dividing the given number by the fourth power of natural numbers starting from 2 (since the number should be a multiple of 2^4).
Let's start by dividing it by 2^4 = 16:
4795795472k / 16 = 299737217k
Now, we divide it by 3^4 = 81:
299737217k / 81 = 3698167k
Next, we divide it by 4^4 = 256:
3698167k / 256 = 14431k
Finally, we divide it by 5^4 = 625:
14431k / 625 = 23.0896k
Since k should be a natural number, the smallest possible value for k is 1. Thus, when k = 1, the product 4795795472k represents a perfect fourth power.
To find the smallest possible natural value for k such that the product 4795795472k represents a perfect fourth power, we need to determine the factors of the given product and check for a perfect fourth power pattern.
First, let's factorize the given product: 4795795472k = 2^10 * 3^2 * 159859849k.
Since we are looking for a perfect fourth power, we need to find a value for k that makes the factorization divisible by 2^15 * 3^4 (because any number raised to the fourth power has its exponent multiplied by 4).
Now, we need to find the smallest value for k that fulfills this condition. To do so, we consider the exponents of the factors:
- For 2: We need the exponent to be divisible by 15, so k needs to be divisible by the difference between 15 and 10, which is 5.
- For 3: We need the exponent to be divisible by 4, so k needs to be divisible by the difference between 4 and 2, which is 2.
To find the smallest natural number that satisfies both conditions, we find the least common multiple (LCM) of the numbers 5 and 2, which is 10. Therefore, the smallest possible natural value for k is 10.
Therefore, if the product 4795795472k represents a perfect fourth power, the smallest possible natural value for k is 10.