What is the least value of n if 194480 + n = m 4, where m and n are natural numbers?
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Understanding the Perfect Fourth Power Problem
The question asks for the smallest natural number 'n' that, when added to 194480, results in a perfect fourth power of another natural number 'm'. In mathematical terms, we are looking for the least natural number 'n' such that \(194480 + n = m^4\), where 'm' is a natural number.
A natural number is a positive integer (1, 2, 3, ...). A perfect fourth power is a number that can be expressed as an integer raised to the power of 4 (e.g., \(1^4=1\), \(2^4=16\), \(3^4=81\), etc.).
Finding the Target Perfect Fourth Power
We need to find a perfect fourth power (\(m^4\)) that is slightly greater than or equal to 194480. This is because 'n' must be a natural number (positive), so \(194480 + n\) must be greater than 194480.
Let's estimate the base 'm' by considering the fourth root of 194480. The fourth root is the square root of the square root.
- We can estimate \(\sqrt{194480}\). Since \(400^2 = 160000\) and \(500^2 = 250000\), the square root is between 400 and 500. \(440^2 = 193600\), which is very close to 194480. So, \(\sqrt{194480}\) is slightly more than 440. Let's say it's around 440-441.
- Now, we need the square root of a number around 440-441. Since \(20^2 = 400\) and \(21^2 = 441\), the square root of 440-441 is around 21.
This suggests that the base 'm' could be around 21.
Calculating Fourth Powers to Find the Least 'n'
Let's calculate the fourth powers of integers close to our estimate, starting with 21:
- For \(m=20\), \(m^4 = 20^4 = (20^2)^2 = 400^2 = 160000\).
\(194480 + n = 160000 \implies n = 160000 - 194480 = -34480\). This is not a natural number, so \(m=20\) is too small. - For \(m=21\), \(m^4 = 21^4 = (21^2)^2\).
\(21^2 = 441\).
\(21^4 = 441^2 = (400 + 41)^2 = 400^2 + 2 \times 400 \times 41 + 41^2 = 160000 + 32800 + 1681 = 194481\).
Now we set \(194480 + n = 194481\).
\(n = 194481 - 194480 = 1\).
Since \(m=21\) gives a perfect fourth power (194481) that is just slightly greater than 194480, the resulting value of \(n=1\) is the smallest possible natural number for this equation to hold.
Let's check the next integer for 'm' to be sure:
- For \(m=22\), \(m^4 = 22^4 = (22^2)^2 = 484^2\).
\(484^2 = (500 - 16)^2 = 500^2 - 2 \times 500 \times 16 + 16^2 = 250000 - 16000 + 256 = 234256\).
\(194480 + n = 234256 \implies n = 234256 - 194480 = 39776\). This is a natural number, but it is much larger than 1.
The smallest perfect fourth power greater than 194480 is 194481, which occurs when \(m=21\). This requires \(n=1\). Any larger perfect fourth power (like 234256 for \(m=22\)) would require a larger value of \(n\). Therefore, the least value of 'n' is 1.
Conclusion on the Least Value of n
We found that for \(m=21\), \(m^4 = 194481\). Setting \(194480 + n = 194481\) yields \(n=1\). Since 1 is the smallest natural number among the calculated possibilities and also the smallest among the given options (1, 2, 3, 4), it is indeed the least value of 'n' that satisfies the condition \(194480 + n = m^4\) for natural numbers 'm' and 'n'.
| Value of m | Value of m4 | Equation: 194480 + n = m4 | Value of n | Is n a natural number? |
|---|---|---|---|---|
| 20 | 160000 | 194480 + n = 160000 | -34480 | No |
| 21 | 194481 | 194480 + n = 194481 | 1 | Yes |
| 22 | 234256 | 194480 + n = 234256 | 39776 | Yes |
The smallest natural number 'n' that makes \(194480 + n\) a perfect fourth power is 1, corresponding to \(m=21\). This matches one of the given options.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Natural Numbers | Positive integers: 1, 2, 3, ... |
| Perfect Fourth Power | A number that is the result of raising a natural number to the power of 4 (\(m^4\)). |
| Least Value of n | The smallest possible natural number 'n' satisfying the given equation. |
Additional Information: Perfect Powers
A perfect power is an integer that can be expressed as \(a^b\), where 'a' is an integer greater than 1, and 'b' is an integer greater than 1. Perfect fourth powers are a specific type of perfect power where the exponent 'b' is 4.
Examples of perfect powers:
- Perfect squares (\(a^2\)): 4, 9, 16, 25, ...
- Perfect cubes (\(a^3\)): 8, 27, 64, 125, ...
- Perfect fourth powers (\(a^4\)): 16, 81, 256, 625, ...
To solve problems like this, where you need to add a minimum value to reach a perfect power, you typically find the smallest perfect power that is greater than the initial number.
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