Final answer:
The smallest natural number value for n that makes the expression √(56n) a natural number is 7, as multiplying 56 by 7 yields 392, which is a perfect square.
Step-by-step explanation:
To determine the smallest natural number value for n that makes the expression √(56n) also a natural number, we need to find a value of n such that 56n is a perfect square. A perfect square is a number that has an integer as its square root.
We know that 56 is 7 multiplied by 8 (56 = 7 × 8). The smallest natural number that we can multiply 56 by to make it a perfect square is the number that completes the square for the prime factor 7. Since 8 is already a perfect square (2²), we only need to multiply 56 by 7 to make 56n a perfect square: 56 × 7 = 392. The square root of 392 is 14², which is a natural number.
Therefore, the smallest natural number value for n is 7.