Question

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Answer 1

1) Find The Rational Number Between -1/2 and 3/4

To find rational numbers between -1/2 and 3/4.

$\sf = \left( - \dfrac{1}{2} + \dfrac{3}{4} \right) \div 2$

$\sf = \left( \dfrac{ - 2 + 3}{4} \right) \div 2$

$\sf = \dfrac{ 1}{4} \times \dfrac{1}{2}$

$\sf = \dfrac{1}{4}$

• Rational numbers between -1/2 and 3/4 Is 1/4.

2) Rationalization

• Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

$\sf \: i) \dfrac{1}{ \sqrt{18} - \sqrt{32} }$

$\sf \: \: = \dfrac{1}{ \sqrt{18} - \sqrt{32} } \times \dfrac{ \sqrt{18} + \sqrt{32} }{ \sqrt{18} + \sqrt{32} }$

$\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{ (\sqrt{18})^{2} + (\sqrt{32}) ^{2} }$

$\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{{18} + {32} }$

$\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{50 }$

$\sf \: ii) \dfrac{2 \sqrt{2} }{ \sqrt{5} + \sqrt{7} }$

$\sf \: = \dfrac{2 \sqrt{2} }{ \sqrt{5} + \sqrt{7} } \times \dfrac{ \sqrt{5 } - \sqrt{7} }{ \sqrt{5} - \sqrt{7} }$

$\sf \: = \dfrac{2 \sqrt{2}( \sqrt{5 } - \sqrt{7} )}{( \sqrt{5}) { }^{2} - ( \sqrt{7} ) {}^{2} }$

$\sf \: = \dfrac{2 \sqrt{2}( \sqrt{5 } - \sqrt{7} )}{5 - 7 }$

$\sf \: = \dfrac{2 \sqrt{10} - 2\sqrt{14} }{ - 2}$

$\sf \: = \dfrac{ - 2( \sqrt{10} - \sqrt{14} ) }{ 2}$

$\sf \: = \dfrac{ \sqrt{14} - \sqrt{10} }{ 2}$

3 Find The value Of x

$\Rightarrow\sf 2 ^{x} = (128)^{ \frac{1}{7} } \times ( \sqrt{2} )^{4}$

$\Rightarrow\sf 2 ^{x} = (2^{7} )^{ \frac{1}{7} } \times ({2} ^{ \frac{1}{2} } )^{4}$

$\Rightarrow\sf 2 ^{x} = (2 )^{ \frac{1}{7} \times 7} \times (2 )^{4 \times \dfrac{1}{2} }$

$\Rightarrow\sf 2 ^{x} = (2 )^{1} \times (2 )^{2}$

• On substituting Both Side

$\sf \: x = 2 + 1$

$\sf \: x = 3$