Math, asked by ghoshpallav27, 8 months ago

3^x = 5^y = 75^z proof that xz/x+y​

Answers

Answered by anindyaadhikari13
6

Correct Question:

  • If \sf 3^{x} = 5^{y} = 75^{z}, prove that, \sf z = \dfrac{xy}{2x+y}

Proof:

 \sf Let \ {3}^{x}  =  {5}^{y} =  {75}^{z}  = k

So,

 \sf  \mapsto{3}^{x}  = k \implies 3 =  {k}^{ \frac{1}{x} }  \: ...(i)

 \sf  \mapsto{5}^{y}  = k \implies 5 =  {k}^{ \frac{1}{y} }  \: ...(ii)

 \sf  \mapsto{75}^{z}  = k \implies 75 =  {k}^{ \frac{1}{z} }  \: ...(iii)

Again,

 \sf \mapsto 75 = 3 \times  {5}^{2}

Substituting the values, we get,

 \sf \mapsto {k}^{ \frac{1}{z} }  =  {k}^{ \frac{1}{x} }  \times ( {k}^{ \frac{1}{y} })^{2}

 \sf \mapsto {k}^{ \frac{1}{z} }  =  {k}^{ \frac{1}{x} }  \times {k}^{ \frac{2}{y} }

 \sf \mapsto {k}^{ \frac{1}{z} }  =  {k}^{ \frac{1}{x} +  \frac{2}{y}  }

Comparing base, we get,

 \sf  \mapsto \small \dfrac{1}{z}  =  \dfrac{1}{x}  +  \dfrac{2}{y}

 \sf  \mapsto \small \dfrac{1}{z}  =  \dfrac{y + 2x}{xy}

 \sf  \mapsto \small \dfrac{1}{z}  =  \dfrac{2x + y}{xy}

 \sf  \mapsto z  =  \dfrac{xy}{2x + y}

(Hence Proved)

Formulae to solve indices questions:

 \sf \mapsto {x}^{0}  = 1,x \neq0

 \sf \mapsto {x}^{a}  \times  {x}^{b}  =  {x}^{a + b}

 \sf  \mapsto({x}^{a} )^{b}  =  {x}^{ab}

 \sf \mapsto \dfrac{ {x}^{a} }{ {x}^{b} }  =  {x}^{a - b} ,x \neq0

 \sf \mapsto {x}^{a}  \times  {y}^{a}  =  {(xy)}^{a}

 \sf \mapsto  \bigg(\dfrac{x}{y} \bigg)^{a}  =  \dfrac{ {x}^{a} }{ {y}^{a} } ,y \neq0

 \sf \mapsto {x}^{a}  =  {x}^{b}  \implies a = b,x \neq0

 \sf \mapsto {x}^{a}  =  {y}^{a}  \implies x = y,a \neq0

 \sf \mapsto  {x}^{a}  =  \dfrac{1}{ {x}^{ - a} }

 \sf \mapsto \sqrt[y]{x}  =  {x}^{ \frac{1}{y} }

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