Proving 2√+5√ is Irrational
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Proving 2√+5√ is Irrational

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Questions and Answers

What is the initial assumption made about 7√?

  • It is a rational number (correct)
  • It is a perfect square
  • It is a whole number
  • It is an irrational number
  • After squaring both sides, what was obtained as the value of 2401?

  • 2401 + 1
  • 2400
  • 2401 * 1 (correct)
  • 2400 + 1
  • What is the conclusion reached about the number 2√+5√?

  • It is an irrational number (correct)
  • It is a rational number
  • It is a fraction
  • It is a whole number
  • What type of numbers are a and b in this context?

    <p>Positive integers</p> Signup and view all the answers

    What technique was used to prove the irrationality of 7√?

    <p>Contradiction</p> Signup and view all the answers

    In the context of real numbers, what does proving irrationality demonstrate?

    <p>&quot;the number cannot be expressed as a fraction or ratio of two integers&quot;</p> Signup and view all the answers

    What is an irrational number?

    <p>A real number that cannot be expressed as a fraction with a finite decimal representation</p> Signup and view all the answers

    Which of the following is an example of an irrational number?

    <p>√5</p> Signup and view all the answers

    What is the expression 2√ + 5√ simplified to?

    <p>7√</p> Signup and view all the answers

    Assuming 7√ is rational, what must be true about a and b in the form a/b?

    <p>a and b must be positive integers</p> Signup and view all the answers

    What property defines an irrational number?

    <p>Cannot be expressed as a fraction with a finite decimal representation</p> Signup and view all the answers

    If 3√ is considered rational, what condition should not be met for the denominator when written as a fraction?

    <p>Not divisible by 7</p> Signup and view all the answers

    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.

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