If find the value of from Mathematics Number Systems

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Number Systems

Short Answer Type

131. Prove that

left parenthesis 3 minus square root of 7 right parenthesis squared

is an irrational number.

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132. Write three numbers whose decimal expansions are non-terminating non-recurring.

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133. Simplify

2 to the power of 2 over 3 end exponent.2 to the power of 1 fifth end exponent

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134. Simplify

open parentheses 1 over 3 cubed close parentheses to the power of 7

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135. Simplify

11 to the power of 1 half end exponent over 11 to the power of 1 fourth end exponent

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136. Simplify

7 to the power of 1 half end exponent.8 to the power of 1 half end exponent

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137. Find the values of a and b if:

fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b.

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138.
If $straight a equals 7 minus 4 square root of 3 comma$ find the value of $square root of straight a plus fraction numerator 1 over denominator square root of a end fraction.$

straight a equals 7 minus 4 square root of 3 equals 4 plus 3 minus 4 square root of 3 equals 2 squared plus left parenthesis square root of 3 right parenthesis squared minus 2 left parenthesis 2 right parenthesis left parenthesis square root of 3 right parenthesis equals space left parenthesis 2 minus square root of 3 right parenthesis squared rightwards double arrow space space space square root of straight a space equals 2 minus square root of 3 space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis space space space space space space fraction numerator 1 over denominator square root of straight a end fraction equals fraction numerator 1 over denominator 2 minus square root of 3 end fraction space space space space space equals space fraction numerator 1 over denominator 2 minus square root of 3 end fraction cross times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction space space space space space equals space fraction numerator 2 plus square root of 3 over denominator 4 minus 3 end fraction space space space space space equals space 2 plus square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis

From (1) and (2),

square root of straight a plus fraction numerator 1 over denominator square root of a end fraction equals left parenthesis 2 minus square root of 3 right parenthesis plus left parenthesis 2 plus square root of 3 right parenthesis equals 4

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139. Simplify:

left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 2 plus square root of 2 right parenthesis squared.

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140. Simplify the following by rationa-lising the denominator:

fraction numerator 7 square root of 3 minus 5 square root of 2 over denominator square root of 48 plus square root of 18 end fraction.

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