To prove that the sum of \( \frac{3}{4} \) and \( \sqrt{10} \) is irrational, we first need to understand the nature of each component.
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Identify the components:
- \( \frac{3}{4} \) is a rational number because it can be expressed as the fraction of two integers (3 and 4).
- \( \sqrt{10} \) is known to be an irrational number. This can be shown by assuming it is rational. If \( \sqrt{10} \) were rational, it could be expressed in the form \( \frac{a}{b} \) (where \( a \) and \( b \) are integers with no common factors, and \( b \neq 0 \)). Squaring both sides gives \( 10 = \frac{a^2}{b^2} \), leading to \( a^2 = 10b^2 \). This implies that \( a^2 \) is even (since the left side is divisible by 10), and consequently, \( a \) must also be even. Let \( a = 2k \) for some integer \( k \). Substituting back, we get \( (2k)^2 = 10b^2 \) or \( 4k^2 = 10b^2 \), which simplifies to \( b^2 = \frac{4k^2}{10} = \frac{2k^2}{5} \). This shows \( b^2 \) also has to be even, hence \( b \) is even. However, this contradicts the original assumption that \( a \) and \( b \) have no common factors since both are even. Therefore, \( \sqrt{10} \) is irrational.
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Combine the numbers:
- Now consider the sum: \( \frac{3}{4} + \sqrt{10} \).
- For the sum of a rational number (like \( \frac{3}{4} \)) and an irrational number (like \( \sqrt{10} \)), we can apply the property that the sum of a rational number and an irrational number is always irrational.
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Conclusion:
- Since \( \frac{3}{4} \) is rational and \( \sqrt{10} \) is irrational, the sum \( \frac{3}{4} + \sqrt{10} \) must be irrational.
In one sentence: The sum of \( \frac{3}{4} \) (a rational number) and \( \sqrt{10} \) (an irrational number) is irrational because the sum of a rational number and an irrational number is always irrational.