Question

{(x^a÷x^b)^c} × {(x^b÷x^c)^a} × {(x^c÷x^a)^b}

Answer 1

ANSWER:

• The value of above expression = 1.

GIVEN:

$(x {}^{a} \div x {}^{b} ) {}^{c} \times (x {}^{b} \div x {}^{c} ) {}^{a} \times (x {}^{c} \div x {}^{a} ) {}^{b}$

TO FIND:

• The value of above expression.

SOLUTION:

$= (x {}^{a} \div x {}^{b} ) {}^{c} \times (x {}^{b} \div x {}^{c} ) {}^{a} \times (x {}^{c} \div x {}^{a} ) {}^{b} \\ \\ = (x {}^{(a - b)} ) {}^{c} \times( x {}^{(b - c)} ) {}^{a} \times( x {}^{(c - a)} ) {}^{b} \\ \\ = x {}^{(a - b)c} \times x {}^{(b - c)a} \times x {}^{(c - a)b } \\ \\ = x {}^{ac - bc} \times x {}^{ab - ac} \times x {}^{bc - ab} \\ \\ = x {}^{ac - bc + ab - ac + bc - ab} \\ \\ = x {}^{0} \\ \\ = 1$

The value of above expression = 1.

NOTE:

• $\implies \: a {}^{x} \times a {}^{y} = a {}^{x + y} \\ \\ \implies \: a {}^{x} \div a {}^{y} = a {}^{x - y} \\ \\ \implies \: (a {}^{x} ) {}^{y} = a {}^{xy}$