Answer:
To find the possible values of x that would result in a whole number simplification of the expression \(\sqrt[x]{441^{2}}\), we need to consider the factors of 441.
1. Prime factorize 441:
- 441 can be written as \(3^2 \times 7^2\).
2. Simplify the expression:
- \(\sqrt[x]{441^{2}}\) can be rewritten as \((3^2 \times 7^2)^{\frac{2}{x}}\).
- Using the properties of exponents, we can simplify further to \(3^{\frac{4}{x}} \times 7^{\frac{4}{x}}\).
3. Determine the values of x for which the exponents result in whole numbers:
- For the exponents to result in whole numbers, the denominator of \(\frac{4}{x}\) must be a factor of 4.
- The possible values for the denominator are 1, 2, and 4, as they are factors of 4.
4. Calculate the corresponding values of x:
- For each possible value of the denominator, we can solve for x by setting the denominator equal to the value and solving the equation: \(\frac{4}{x} = \text{denominator}\).
- For a denominator of 1, \(x = 4\).
- For a denominator of 2, \(x = 2\).
- For a denominator of 4, \(x = 1\).
Therefore, the possible values of x that would result in a whole number simplification of \(\sqrt[x]{441^{2}}\) are 4, 2, and 1.
Explanation:
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