Question

If 5^x = 3^y =45^z, prove that
1/z = 1/x + 2/y


Please answer....
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Answer 1

Answer:

1 / z = 2/y + 1/x

Explanation:

let 5^x = 3^y = 45^z = k

 5^x = k   ⇒ 5 = k^(1/x)

 3^y = k   ⇒ 3 = k^(1/y)

45^z = k   ⇒ 45 = k^(1/z)

⇒ 45 = 3 * 3 * 5

⇒ 45 = 3^2 * 5

⇒ k^(1/z) = [ k^(1/y) ]^2 * k^(1/x)

⇒ k^(1/z) = k^(2/y) * k^(1/x)

⇒ k^(1/z) = k^( 2/y + 1/x )

⇒ 1/z = 2/y + 1/x

    Proved.

Answer 2

•5^x=3^y=45^z

By squaring 45^z ,

(45^z)^2 = (5^x)×(35^y)

Expanding 45,

5×9=45

=(5×3^2)^2z = (5^x)×(35^y)

=5^2z×3^4z = (5^x)×(35^y)

Hence,

2z =x and 4z = y

1/x = 1/2z,

1/y = 1/4z

Hence,

1/x + 2/y = 1/2z + 1/2z = 1/z

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