The product of a multiplication problem being equal to \( \sqrt{255} \) implies that the factors involved in this multiplication must be such that their product equals \( \sqrt{255} \).
Understanding \( \sqrt{255} \)
First, let's simplify \( \sqrt{255} \). The number 255 can be factored as follows:
\[ 255 = 3 \times 5 \times 17 \]
Thus, we can express \( \sqrt{255} \) as:
\[ \sqrt{255} = \sqrt{3 \times 5 \times 17} \]
Factors of \( \sqrt{255} \)
Since \( \sqrt{255} \) is not a perfect square, we can infer the following about its factors:
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Irrational Factors: Any factors of \( \sqrt{255} \) will likely be irrational numbers unless we specifically choose numbers that, when multiplied, give \( \sqrt{255} \).
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Example Factors: You could express \( \sqrt{255} \) as the product of two factors in various ways, such as:
- \( a = \sqrt{3} \) and \( b = \sqrt{85} \), where \( \sqrt{3} \times \sqrt{85} = \sqrt{15} \)
- \( a = 1 \) and \( b = \sqrt{255} \)
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Non-Integer: Since \( \sqrt{255} \) is not an integer, at least one of the factors must be irrational unless both are chosen to be specific ratios (e.g., fractions).
Conclusion
In summary, the factors that multiply together to give \( \sqrt{255} \) can be expressed as irrational numbers, and at least one of the factors cannot be expressed as a simple integer. You can choose a multitude of pairs of factors that will satisfy this multiplication problem, but they will generally involve at least one irrational component.
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