Question

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Answer 1

Refer the attachment for answer.

Answer 2

Question:

Find the value of x if

$\displaystyle{\sf\:(\:49\:)^{x\:+\:4}\:=\:7^2\:\times\:(\:343\:)^{x\:+\:1}}$

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:3\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:(\:49\:)^{x\:+\:4}\:=\:7^2\:\times\:(\:343\:)^{x\:+\:1}}$

We have to find the value of x.

Now,

$\displaystyle{\sf\:(\:49\:)^{x\:+\:4}\:=\:7^2\:\times\:(\:343\:)^{x\:+\:1}}$

$\displaystyle{\implies\sf\:(\:7^2\:)^{x\:+\:4}\:=\:7^2\:\times\:(\:7^3\:)^{x\:+\:1}}$

$\displaystyle{\implies\sf\:7^{2\:(\:x\:+\:4\:)}\:=\:7^2\:\times\:7^{3\:(\:x\:+\:1\:)}\:\qquad\dots\:[\:(\:a^m\:)^n\:=\:a^{mn}\:]}$

$\displaystyle{\implies\sf\:7^{2x\:+\:8}\:=\:7^2\:\times\:7^{3x\:+\:3}}$

$\displaystyle{\implies\sf\:7^{2x\:+\:8}\:=\:7^{2\:+\:3x\:+\:3}\:\qquad\dots\:[\:a^m\:\times\:a^n\:=\:a^{m\:+\:n}\:]}$

$\displaystyle{\implies\sf\:7^{2x\:+\:8}\:=\:7^{3x\:+\:5}}$

$\displaystyle{\implies\sf\:2x\:+\:8\:=\:3x\:+\:5\:\qquad\dots\:[\:Bases\:are\:equal\:]}$

$\displaystyle{\implies\sf\:8\:-\:5\:=\:3x\:-\:2x}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:x\:=\:3\:}}}}$