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Prove that: ((x^a)/(x^b))^c\xx\ ((x^b)/(...

Prove that: `((x^a)/(x^b))^c\xx\ ((x^b)/(x^c))^a\ xx\ ((x^c)/(x^a))^b=1`

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To prove the expression \[ \left(\frac{x^a}{x^b}\right)^c \cdot \left(\frac{x^b}{x^c}\right)^a \cdot \left(\frac{x^c}{x^a}\right)^b = 1, \] we will simplify the left-hand side (LHS) step by step. ...
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RD SHARMA-EXPONENTS OF REAL NUMBER-All Questions
  1. If a=3 and b=-2, find the values of: (i)\ a^a+b^b (ii)\ a^b+b^a (iii...

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  2. Prove that: ((x^a)/(x^b))^(a^(2+a b+b^(2)))xx\ ((x^b)/(x^c))^(b^(2+b c...

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  3. Prove that: ((x^a)/(x^b))^c\xx\ ((x^b)/(x^c))^a\ xx\ ((x^c)/(x^a))^b=1

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  4. Prove that: 1/(1+x^(a-b))+1/(1+x^(b-a))=1

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  5. Prove that: 1/(1+x^(b-a)+\ x^(c-a))+1/(1+x^(a-b)+\ x^(c-b))+1/(1+x^(b-...

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  6. Prove that: (a+b+c)/(a^(-1)\ b^(-1)+b^(-1)\ c^(-1)+c^(-1)a^(-1))=abc

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  7. Prove that: (a^(-1)+b^(-1))^(-1)=(ab)/(a+b)

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  8. If a b c=1,\ show that 1/(1+a+b^(-1))+1/(1+b+c)+1/(1+c+a^(-1))=1

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  9. Simplify the following: (i)\ (3^n\ xx\ 9^(n+1))/(3^(n-1)\ xx\ 9^(n-1))...

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  10. Simplify the following: (i)\ (5^(n+3)-\ 6\ xx\ 5^(n+1))/(9\ xx\ 5^n-2^...

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  11. Solve the following equations for x (i)7^(2x+3)=1 (ii)2^(x+1)=4^x (...

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  12. Solve the following equations for x\ : (i)\ 4^(2x)=1/(32)\ (ii)4^(x-1...

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  13. Solve the following equations for x\ : (i)\ 2^(2x)-\ 2^(x+3)+2^4=0 ...

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  14. If 49392=a^4b^2c^3, find the value of a ,\ b\ a n d\ c , where a ,\ b\...

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  15. If 1176=2^a xx\ 3^b xx\ 7^c , and a , b\ a n d\ c are natural numbers...

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  16. Given 4725=3^a5^b7^c , find the integral values of a ,\ b\ a n d\ c t...

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  17. If a=x y^(p-1),\ b=x y^(q-1) and c=x y^(r-1) , prove that a^(q-r)b^(r...

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  18. Find the value of: (8/(27))^(1/3)

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  19. Find the value of : ((32)/(243))^(1/5)

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  20. Find the value of: (1/(64))^(1/6)

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