simplify the complex fraction. \n\\( \\frac { \\frac { 1 } { x } + \\frac { 4 } { x ^ { 2 } } } { x + \\frac…

simplify the complex fraction. \n\\( \\frac { \\frac { 1 } { x } + \\frac { 4 } { x ^ { 2 } } } { x + \\frac { 64 } { x ^ { 2 } } } \\)\n\\( \\frac { \\frac { 1 } { x } + \\frac { 4 } { x ^ { 2 } } } { x + \\frac { 64 } { x ^ { 2 } } } = \\square \\)

simplify the complex fraction. \n\\( \\frac { \\frac { 1 } { x } + \\frac { 4 } { x ^ { 2 } } } { x + \\frac { 64 } { x ^ { 2 } } } \\)\n\\( \\frac { \\frac { 1 } { x } + \\frac { 4 } { x ^ { 2 } } } { x + \\frac { 64 } { x ^ { 2 } } } = \\square \\)

Answer

Explanation:

Step1: Simplify numerator and denominator

For the numerator (\frac{1}{x}+\frac{4}{x^{2}}), find a common - denominator (x^{2}). Then (\frac{1}{x}+\frac{4}{x^{2}}=\frac{x + 4}{x^{2}}). For the denominator (x+\frac{64}{x^{2}}), find a common - denominator (x^{2}). Then (x+\frac{64}{x^{2}}=\frac{x^{3}+64}{x^{2}}). So the complex fraction becomes (\frac{\frac{x + 4}{x^{2}}}{\frac{x^{3}+64}{x^{2}}}).

Step2: Divide the two fractions

When dividing by a fraction, multiply by its reciprocal. So (\frac{\frac{x + 4}{x^{2}}}{\frac{x^{3}+64}{x^{2}}}=\frac{x + 4}{x^{2}}\times\frac{x^{2}}{x^{3}+64}). Use the sum - of - cubes formula (a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})), where (a=x) and (b = 4) (since (x^{3}+64=x^{3}+4^{3})). Then (x^{3}+64=(x + 4)(x^{2}-4x + 16)). So (\frac{x + 4}{x^{2}}\times\frac{x^{2}}{(x + 4)(x^{2}-4x + 16)}).

Step3: Cancel out common factors

Cancel out the common factors (x + 4) and (x^{2}).

Answer:

(\frac{1}{x^{2}-4x + 16})