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1) Find The Rational Number Between -1/2 and 3/4
To find rational numbers between -1/2 and 3/4.
[tex] \sf = \left( - \dfrac{1}{2} + \dfrac{3}{4} \right) \div 2[/tex]
[tex] \sf = \left( \dfrac{ - 2 + 3}{4} \right) \div 2[/tex]
[tex]\sf = \dfrac{ 1}{4} \times \dfrac{1}{2} [/tex]
[tex] \sf = \dfrac{1}{4} [/tex]
2) Rationalization
[tex] \sf \: i) \dfrac{1}{ \sqrt{18} - \sqrt{32} } [/tex]
[tex]\sf \: \: = \dfrac{1}{ \sqrt{18} - \sqrt{32} } \times \dfrac{ \sqrt{18} + \sqrt{32} }{ \sqrt{18} + \sqrt{32} } [/tex]
[tex]\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{ (\sqrt{18})^{2} + (\sqrt{32}) ^{2} }[/tex]
[tex]\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{{18} + {32} }[/tex]
[tex]\sf \: \: = \dfrac{ \sqrt{18} + \sqrt{32} }{50 }[/tex]
[tex] \sf \: ii) \dfrac{2 \sqrt{2} }{ \sqrt{5} + \sqrt{7} } [/tex]
[tex] \sf \: = \dfrac{2 \sqrt{2} }{ \sqrt{5} + \sqrt{7} } \times \dfrac{ \sqrt{5 } - \sqrt{7} }{ \sqrt{5} - \sqrt{7} } [/tex]
[tex] \sf \: = \dfrac{2 \sqrt{2}( \sqrt{5 } - \sqrt{7} )}{( \sqrt{5}) { }^{2} - ( \sqrt{7} ) {}^{2} } [/tex]
[tex] \sf \: = \dfrac{2 \sqrt{2}( \sqrt{5 } - \sqrt{7} )}{5 - 7 } [/tex]
[tex]\sf \: = \dfrac{2 \sqrt{10} - 2\sqrt{14} }{ - 2} [/tex]
[tex]\sf \: = \dfrac{ - 2( \sqrt{10} - \sqrt{14} ) }{ 2} [/tex]
[tex]\sf \: = \dfrac{ \sqrt{14} - \sqrt{10} }{ 2} [/tex]
3 Find The value Of x
[tex] \Rightarrow\sf 2 ^{x} = (128)^{ \frac{1}{7} } \times ( \sqrt{2} )^{4} [/tex]
[tex] \Rightarrow\sf 2 ^{x} = (2^{7} )^{ \frac{1}{7} } \times ({2} ^{ \frac{1}{2} } )^{4} [/tex]
[tex]\Rightarrow\sf 2 ^{x} = (2 )^{ \frac{1}{7} \times 7} \times (2 )^{4 \times \dfrac{1}{2} } [/tex]
[tex]\Rightarrow\sf 2 ^{x} = (2 )^{1} \times (2 )^{2} [/tex]
[tex] \sf \: x = 2 + 1[/tex]
[tex]\sf \: x = 3[/tex]