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Number Systems

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131. 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 equals space 9 plus 7 minus 2.3. square root of 7 equals space 16 minus 6 square root of 7

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

is an irrational number as 6 is a non-zero rational number and  i

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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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 $square root of straight a plus fraction numerator 1 over denominator square root of a end fraction.$

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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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Previous Year Papers

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Test Yourself

Short Answer Type

131. Prove that is an irrational number.

131. Prove that $left parenthesis 3 minus square root of 7 right parenthesis squared$ is an irrational number.