simplify.\n\\( \\frac { x ^ { - 1 } } { x ^ { - 2 } + y ^ { - 6 } } \\)\n\\( \\frac { x ^ { - 1 } } { x ^ {…

simplify.\n\\( \\frac { x ^ { - 1 } } { x ^ { - 2 } + y ^ { - 6 } } \\)\n\\( \\frac { x ^ { - 1 } } { x ^ { - 2 } + y ^ { - 6 } } = \\)

simplify.\n\\( \\frac { x ^ { - 1 } } { x ^ { - 2 } + y ^ { - 6 } } \\)\n\\( \\frac { x ^ { - 1 } } { x ^ { - 2 } + y ^ { - 6 } } = \\)

Answer

Explanation:

Step1: Use the negative exponent rule (a^{-n}=\frac{1}{a^n})

The expression (\frac{x^{-1}}{x^{-2}+y^{-6}}) can be rewritten as (\frac{\frac{1}{x}}{\frac{1}{x^{2}}+\frac{1}{y^{6}}}).

Step2: Find a common denominator for the denominator

The common denominator of (x^{2}) and (y^{6}) is (x^{2}y^{6}). So (\frac{1}{x^{2}}+\frac{1}{y^{6}}=\frac{y^{6}+x^{2}}{x^{2}y^{6}}).

Step3: Rewrite the complex - fraction

The complex - fraction (\frac{\frac{1}{x}}{\frac{y^{6}+x^{2}}{x^{2}y^{6}}}) is equivalent to (\frac{1}{x}\times\frac{x^{2}y^{6}}{x^{2}+y^{6}}) (using the rule (\frac{a/b}{c/d}=\frac{a}{b}\times\frac{d}{c})).

Step4: Simplify the expression

(\frac{1}{x}\times\frac{x^{2}y^{6}}{x^{2}+y^{6}}=\frac{xy^{6}}{x^{2}+y^{6}}) (by canceling out one (x) from the numerator and denominator).

Answer:

(\frac{xy^{6}}{x^{2}+y^{6}})