Tag: proofs of irrationality

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Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

$\sqrt{5}\left{(\sqrt{5}+1)^{50}-(\sqrt{5}-1)^{50}\right}$ is?

  1. An irrational number
  2. $0$

  3. A natural number

  4. A prime number
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This expression involves powers of (sqrt(5)+1) and (sqrt(5)-1). Using the binomial expansion, the irrational parts cancel out, leaving an irrational result due to the leading sqrt(5).

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

Find x if $\dfrac{\sqrt{3x+1}+\sqrt{3x-6}}{\sqrt{3x+1}-\sqrt{3x-6}}=7$.

  1. $2$
  2. $5$
  3. $3$
  4. $7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\dfrac { \sqrt { 3x+1 } +\sqrt { 3x-6 }  }{ \sqrt { 3x+1 } -\sqrt { 3x-6 }  } =7$

Rotational give :-

$\dfrac { \left( 3x+1 \right) +\left( 3x-6 \right) +2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  }  }{ \left( 3x+1 \right) -\left( 3x-6 \right)  } =7$

$\Rightarrow 6x-5+2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =49$

$\Rightarrow 2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =-6x+54$

$\Rightarrow \sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =-3x+27$
which gives, $x=5$
Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

The simplified form of the expression $\sqrt { \sqrt [ 3 ]{ 729{ x }^{ 12 } }  } -\dfrac { { x }^{ -2 }-{ x }^{ -3 } }{ { x }^{ -4 }-{ x }^{ -5 } } $ is

  1. ${ 3x }^{ 2 }$
  2. ${ 3x }^{ 3 }$
  3. ${ 2x }^{ 2 }$
  4. ${ 4x }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt{(729x^{12})^{\frac{1}{3}}}-\cfrac{x^{-2}-x^{-3}}{x^{-4}-x^{-5}}$


$=\sqrt{(3^6x^{12})^{\frac{1}{3}}}-\cfrac{x^{-2}-x^{-3}}{x^{-4}-x^{-5}}$


$=\sqrt{(3^2x^4)}-\cfrac{x^{-2}(1-x^{-1})}{x^{-4}(1-x^{-1})}$

$=3x^2-x^2$

$\=2x^2$