Step-by-step explanation:
[tex] \underline{ \underline{ \text{Given}}}[/tex] :
- a = dk³ , b = dk² , c = dk
[tex] \underline{ \underline{ \text{To \: prove}}} : [/tex]
[tex] \tt{ \sqrt{ab} - \sqrt{bc} - \sqrt{cd} = \sqrt{(a - b - c)(b - c - d)} } [/tex]
[tex] \underline{ \underline{ \text{Solution}}} : [/tex]
Left hand side ( L.H.S ) :
[tex] \tt{ \sqrt{ab} - \sqrt{bc} - \sqrt{cd}} [/tex]
⟶ [tex] \tt{ \sqrt{d {k}^{3} \cdot \: d {k}^{2} } - \sqrt{d {k}^{2} \cdot \: dk } - \sqrt{dk \cdot \: d} }[/tex]
⟶ [tex] \tt{ \sqrt{ {d}^{2} {k}^{5} } - \sqrt{ {d}^{2} {k}^{3} } - \sqrt{ {d}^{2}k } }[/tex]
⟶ [tex] \tt{ \sqrt{ {d}^{2} \cdot \: {k}^{2} \cdot \: {k}^{2} \cdot \: k } - \sqrt{ {d}^{2} \cdot \: {k}^{2} \cdot \: k \: } - \sqrt{ {d}^{2}k } }[/tex]
⟶ [tex] \tt{ \sqrt{ {d}^{2} } \cdot \sqrt{ {k}^{2} } \cdot \: \sqrt{ {k}^{2} } \cdot \: \sqrt{k} } - \sqrt{ {d}^{2} }\cdot \: \sqrt{ {k}^{2}} \cdot \: \sqrt{k} \: - \sqrt{ {d}^{2} } \cdot \: \sqrt{k} [/tex]
⟶ [tex] \tt{d {k}^{2} \sqrt{k} - dk \sqrt{k} - d \sqrt{k} }[/tex]
⟶ [tex] \tt{ d\sqrt{k} \: ( {k}^{2} - k - 1) } [/tex]
Right Hand Side ( R.H.S) :
[tex] \tt{ \sqrt{(a - b - c)(b - c - d)}} [/tex]
⟶ [tex] \tt{ \sqrt{(d {k}^{3} - d {k}^{2} - dk)( d{k}^{2} - dk - d)} }[/tex]
⟶ [tex] \tt{ \sqrt{ \{dk( {k}^{2} - k - 1) \} \: \{d( {k}^{2} - k - 1) \}} }[/tex]
⟶ [tex] \tt{ \sqrt{ {d}^{2}k( {k}^{2} - k - 1) ^{2} } } [/tex]
⟶ [tex] \tt{ \sqrt{ {d}^{2}k } \cdot \: \sqrt{ {( {k}^{2} - k - 1)}^{2} } } [/tex]
⟶[tex] \tt{ \sqrt{ {d}^{2} } \cdot \: \sqrt{k} \cdot \: \sqrt{( {k}^{2} - k - 1) ^{2} } }[/tex]
⟶ [tex] \tt{d \sqrt{k} \: ( {k}^{2} - k - 1) } [/tex]
L.H.S = R.H.S
Hence , Proved !
Hope I helped ! ♡
Have a wonderful day / night ! ツ
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