Simplify the following into a fraction with rational denominator

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Number Systems

Short 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 $fraction numerator 1 over denominator square root of 5 plus square root of 6 minus 11 end fraction.$

fraction numerator 1 over denominator square root of 5 plus square root of 6 minus square root of 11 end fraction equals space fraction numerator 1 over denominator square root of 5 plus square root of 6 minus square root of 11 end fraction space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator square root of 5 plus square root of 6 plus square root of 11 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator left parenthesis square root of 5 plus square root of 6 right parenthesis squared minus left parenthesis square root of 11 right parenthesis squared end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 5 plus 6 plus 2 square root of 30 minus 11 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 2 square root of 30 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 2 square root of 30 end fraction fraction numerator square root of 30 over denominator square root of 30 end fraction equals space fraction numerator square root of 150 plus square root of 180 plus square root of 330 over denominator 60 end fraction equals space fraction numerator 5 square root of 6 plus 6 square root of 5 plus square root of 330 over denominator 60 end fraction

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149. Prove that

left parenthesis 3 minus square root of 7 right parenthesis squared

is an irrational number.

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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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