Question

Evaluate for x under root 5 upon 3 the whole raised to x minus 8 equals to 27 upon 125 the whole raised to 2x minus 3

Answer 1

Correct Question:

Solve for x. If $\sf{\bigg(\sqrt{ \dfrac{5}{3} } \bigg)^{x \: - \: 8} \:=\: \bigg(\dfrac{27}{125} \bigg)^{2x\:-\:3}}$

Solution:

$\sf{\bigg(\sqrt{ \dfrac{5}{3} } \bigg)^{x \: - \: 8} \:=\: \bigg(\dfrac{27}{125} \bigg)^{2x\:-\:3}}$

We can write 27 as 3³ as 3 × 3 × 3 = 27 and 125 as 5³ as 5 × 5 × 5 = 125

And √ (under root) as 1/2

$\implies\:\sf{\bigg({ \dfrac{5}{3} } \bigg)^{ \frac{1}{2}^{(x - 8)} }\:=\: \bigg[\dfrac{ {(3)}^{3} }{ {(5)}^{3} } \bigg]^{2x\:-\:3}}$

Resolve the LHS

$\implies\:\sf{\bigg({ \dfrac{3}{5} } \bigg)^{ \frac{-1}{2}^{(x - 8)} }\:=\: \bigg[\dfrac{ {(3)}^{3} }{ {(5)}^{3} } \bigg]^{2x\:-\:3}}$

Now take 3 as common power

$\implies\:\sf{ \bigg(\dfrac{3}{5} \bigg)^{\frac{-1}{2}^{(x\:-\:8)}}\:=\: \bigg(\dfrac{ {3} }{ {5}} \bigg)^{3(2x\:-\:3)}}$

Solve the power

$\implies\:\sf{ \bigg(\dfrac{3}{5} \bigg)^{\frac{-1}{2}^{x\:-\:8}}\:=\: \bigg(\dfrac{ {3} }{ {5}} \bigg)^{6x\:-\:9}}$

3/5 is common on both sides

On comparing we get,

$\implies\:\sf{\dfrac{-1}{2}(x\:-\:8)\:=\:6x\:-\:9}$

$\implies\:\sf{x\:-\:8\:=\:-2(6x\:-\:9)}$

$\implies\:\sf{x\:-\:8\:=\:-12x\:+\:18}$

$\implies\:\sf{x\:+\:12x\:=\:+\:18\:+\:8}$

$\implies\:\sf{13x\:=\:26}$

$\implies\:\sf{x\:=\:\dfrac{26}{13}}$

$\implies\:\sf{x\:=\:2}$

Answer 2

Step-by-step explanation:

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