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Questions and Answers
What is the initial assumption made about 7√?
After squaring both sides, what was obtained as the value of 2401?
What is the conclusion reached about the number 2√+5√?
What type of numbers are a and b in this context?
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What technique was used to prove the irrationality of 7√?
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In the context of real numbers, what does proving irrationality demonstrate?
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What is an irrational number?
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Which of the following is an example of an irrational number?
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What is the expression 2√ + 5√ simplified to?
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Assuming 7√ is rational, what must be true about a and b in the form a/b?
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What property defines an irrational number?
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If 3√ is considered rational, what condition should not be met for the denominator when written as a fraction?
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Study Notes
Proving 2√+5√ is Irrational
Understanding irrational numbers and their properties is a fundamental aspect of algebra and real numbers. To establish that a particular expression is irrational, we must explore its structure and the underlying definitions. In this case, we're looking at the expression 2√+5√.
Irrational Numbers
By definition, an irrational number is a real number that cannot be expressed as a fraction (rational number) with a finite or repeating decimal representation. This includes numbers like π (pi), √2, and √7, which cannot be expressed as fractions with a finite number of digits.
Proving Irrationality
To prove that 2√+5√ is irrational, we'll show that it cannot be expressed as a fraction with a finite or repeating decimal representation. We can start by expanding the expression:
2√ + 5√ = (2 + 5)√ = 7√
Now, we'll assume that 7√ is a rational number, which means it can be expressed as a fraction with a finite or repeating decimal representation. If 7√ is rational, then we can write it as:
7√ = a/b
where a and b are positive integers and b is not divisible by 7 (to avoid 7 as a common factor).
Next, we'll square both sides:
(7√)^2 = (a/b)^2
Now, expand the left side:
7^2 * √^2 = a^2/b^2
Since 7^2 = 49, we have:
49 * √ = a^2/b^2
Now, simplify the expression:
49√ = a^2/b^2
Now, square both sides again to eliminate the square root:
(49√)^2 = (a^2/b^2)^2
49^2 * √^2 = a^4/b^4
Since 49^2 = 2401, we have:
2401 * 1 = a^4/b^4
2401 = a^4/b^4
Now, we'll use the fact that a and b are positive integers and not divisible by 7. Since 2401 cannot be expressed as a perfect square of an integer, the equation is not satisfied. Therefore, our initial assumption that 7√ is a rational number is false.
Conclusion
We've shown that 7√ cannot be expressed as a rational number, and by extension, neither can 2√+5√, since it is equal to 7√. This demonstrates that 2√+5√ is irrational.
This process of contradiction, expanding an expression, and reaching a conclusion is a common technique for proving irrationality. It's a powerful tool to understand and prove properties of real numbers.
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Description
Explore the process of proving the sum of 2√ and 5√, which equals 7√, is an irrational number. Understand the definition of irrational numbers and utilize proof by contradiction to demonstrate that the expression cannot be expressed as a rational number.