# If Abc = 1, Show that 1/(1+A+B^-1)+1/(1+B+C^-1)+1/(1+C+A^-1)=1 - Mathematics

#### Question

If abc = 1, show that 1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1

#### Solution

Consider the left hand side:

1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)

=1/(1+a+1/b)+1/(1+b+1/c)+1/(1+c+1/a)

=1/((b+ab+1)/b)+1/((c+bc+1)/c)+1/((a+ac+1)/a)

=b/(b+ab+1)+c/(c+bc+1)+a/(a+ac+1)        ...........(1)

We know that abc = 1

therefore c = 1/(ab)

By substituting the value of c in equation (1), we get

=b/(b+ab+1)+(1/(ab))/(1/(ab)+b(1/(ab))+1)+a/(a+a(1/(ab))+1)

=b/(b+ab+1)+(1/(ab))/(1/(ab)+b/(ab)+(ab)/(ab))+a/((ab)/b+1/b+b/b)

=b/(b+ab+1)+(1/(ab))/((1+b+ab)/(ab))+a/((ab+1+b)/(b))

=b/(b+ab+1)+(1/(ab)xxab)/(1+b+ab)+(axxb)/(ab+1+b)

=b/(b+ab+1)+1/(b+ab+1)+(ab)/(b+ab+1)

=(b+ab+1)/(b+ab+1)

= 1

Therefore, LHS = RHS

Hence, proved

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#### APPEARS IN

RD Sharma Solution for Mathematics for Class 9 (2018 (Latest))
Chapter 2: Exponents of Real Numbers
2.1 | Q: 6 | Page no. 12

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If Abc = 1, Show that 1/(1+A+B^-1)+1/(1+B+C^-1)+1/(1+C+A^-1)=1 Concept: Laws of Exponents for Real Numbers.