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Questions and Answers
What is the initial assumption made about 7√?
What is the initial assumption made about 7√?
After squaring both sides, what was obtained as the value of 2401?
After squaring both sides, what was obtained as the value of 2401?
What is the conclusion reached about the number 2√+5√?
What is the conclusion reached about the number 2√+5√?
What type of numbers are a and b in this context?
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√?
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?
In the context of real numbers, what does proving irrationality demonstrate?
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What is an irrational number?
What is an irrational number?
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Which of the following is an example of an irrational number?
Which of the following is an example of an irrational number?
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What is the expression 2√ + 5√ simplified to?
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?
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?
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?
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.