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If x^y=y^x, then prove that (x/y)^(x/y)=...

If `x^y=y^x`, then prove that `(x/y)^(x/y)=(x)^((x)/(y)-1)`.

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The correct Answer is:
`(x)^((x)/(y)-1)`
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CALCUTTA BOOK HOUSE-LAWS OF INDICES-EXERCISE - 6
  1. If a^x=b,b^y =c,c^z=a, find the value of xyz.

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  2. Calculate : 9^(-3)xx((16)^(1/4))/(6^(-2))xx((1)/(27))^(-4/3).

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  3. If x^y=y^x, then prove that (x/y)^(x/y)=(x)^((x)/(y)-1).

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  4. Prove that (a^(3/2)+ab)/(ab-b^3)-(sqrt(a))/(sqrt(a)-b)=(sqrt(a))/(b).

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  5. Show that (x^(b-c))^((1)/(bc))xx(x^(c-a))^((1)/(ca))xx(x^(a-b))^((1)/(...

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  6. Factorise : (a+b).

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  7. Find the value of ((bc)^(b-c)(ca)^(c-a)(ab)^(a-b))/((a^(b-c)b^(c-a)c^(...

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  8. If p=a^x,q=a^y" and "a^2=(p^yq^x)^z, then find the value of xyz.

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  9. If a^(m^n)= (a^m)^n, then express m in terms of n.

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  10. If a^m a^n =a^(mn), then express m in terms of n.

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  11. If x=2,y=4, then what is the value of x+2y+sqrt((x-2y)^2)?

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  12. Which one is the greatest ? 3^(2^(2^2)), 3^(2.2.2), 3.2.2.2.

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  13. Find the value of root(3)(x^4sqrt(x^(-5)sqrt(x^6))).

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  14. Calculate : ((1)/(81))^(-5/4).

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  15. Calculate : {root(3)(4)xx(1)/(root(6)(8))xxroot(12)(2^(-1))}^(3/4).

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  16. Simplify : x^(1/3).x^(-1/4).x^(-1/2) divx^(-(7)/(12))

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  17. Simplify : root(4)(a^2b^3c^4)xxroot(3)(a^4b^6)xx(a^(1/2)b^(1/3)c^(1/4)...

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  18. Simplify : {(a)^(n^(2)-1)}^((n)/(n+1))xx(sqrt(a^(2n)))/(a).

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  19. Simplify : ((p+(1)/(q))^m(p-(1)/(q))^n)/((q+(1)/(p))^m(q-(1)/(p))^n).

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  20. Simplify : (2^n.6^(m+1).10^(m-n).15^(m+n-2))/(4^m.3^(m+n).25^(m-1)).

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