Prove that  is an irrational number. from Mathematics Number
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    Number Systems

    Multiple Choice QuestionsShort Answer Type

    141.

    If  find the value of straight x squared plus 1 over straight x squared.

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    142. Let ‘a’ he a rational number and ‘b’ be an irrational number. Is ‘ab’ necessarily an irrational? Justify your answer with an example.
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    143.

    If straight a equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction space and space straight b space equals fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction, find a2 - b2.


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    144.

    If straight x equals 3 plus 2 square root of 2 comma check whether straight x plus 1 over straight x is rational or irrational.

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    145.

    If fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals straight a square root of 3 plus straight b square root of 5 comma find a and b.

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    146. Simplify the following by rationalising the denominators:

    fraction numerator 2 over denominator square root of 5 plus square root of 3 end fraction plus fraction numerator 1 over denominator square root of 2 plus square root of 3 end fraction minus fraction numerator 3 over denominator square root of 5 plus square root of 2 end fraction
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    147. Evaluate: fraction numerator square root of 5 plus square root of 2 over denominator square root of 5 minus square root of 2 end fraction comma given that square root of 10 equals 3.162.
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    148. Simplify the following into a fraction with rational denominator fraction numerator 1 over denominator square root of 5 plus square root of 6 minus 11 end fraction.
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    149. Prove that left parenthesis 3 minus square root of 7 right parenthesis squared is an irrational number.

    left parenthesis 3 minus square root of 7 right parenthesis squared space space space space space space equals space 9 plus 7 minus 2.3. square root of 7 space space space space space space space equals 16 minus 6 square root of 7 space space space space space space

    6 square root of 7 is an irrational number as 6 is a non-zero rational number and

    square root of 7 is an irrational number and the product of a non-zero rational number with an irrational number is irrational.

    Again, 16 is a rational number and 6 square root of 7 is an irrational number. So, 16 minus 6 square root of 7 is an irrational number since the difference of a rational number and an irrational number is irrational.
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    150. Evaluate cube root of left parenthesis 343 right parenthesis to the power of negative 2 end exponent end root space space
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