Prove that (3+25) is an irrational number, given that 5 is an irrational number.
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Step-by-Step Solution
Step 1: Assume the opposite
To prove that 3+25 is an irrational number, we will use the method of contradiction. We assume, for the sake of contradiction, that 3+25 is a rational number. If it is rational, it can be expressed as a fraction ba, where a and b are integers, b=0, and a, b are coprime (have no common factors other than 1).
Step 2: Isolate the irrational term
Now, we rearrange the equation to isolate the term involving 5. We subtract 3 from both sides of the equation.
Step 3: Simplify the rational side
We combine the terms on the right-hand side by finding a common denominator. This simplifies the expression to a single fraction.
Step 4: Further isolate the irrational term
To completely isolate 5, we divide both sides of the equation by 2. This gives us an expression for 5 in terms of a and b.
Step 5: Reach a contradiction
Since a and b are integers, a - 3b will also be an integer, and 2b will be a non-zero integer. Therefore, the expression 2ba−3b is a rational number. This implies that 5 is a rational number. However, we are given that 5 is an irrational number. This creates a contradiction, as an irrational number cannot be equal to a rational number.
Step 6: Conclusion
Our initial assumption that 3+25 is rational led to a contradiction. Therefore, our assumption must be false, and 3+25 must be an irrational number.