Question

3^x=5^y=15^z, show that z(x+y)=xy​

Answer 1

Answer:

Let 3^x=5^y=15^z=k

3=k¹/x,5=k¹/y,15=k¹/z

Now 15=3×5

k1/z = k¹/x × k¹/y

k¹/z = k¹/x+¹/y

1/z = x+y/(xy)

z = (xy)/x+y

z(x+y) = xy

hence proved

hope u understand

Answer 2

Let 3^x = 5^y = 15^z = k

3^x = k

3 = k^( 1/x )

5^y = k

5 = k^( 1/y )

15^z = k

15 = k^( 1/z )

We know that

3 × 5 = 15

Substituting the values

k^( 1/x ) × k^( 1/y ) = k^( 1/z )

Using laws of exponents a^m × a^n = a^( m + n)

k^( 1/x + 1/y ) = k^( 1/z )

If bases are equal their exponents also will be equal

1/x + 1/y = 1/z

( x + y)/xy = 1/z

z(x + y) = xy

Hence shown.