Question

Simplify each complex fraction. [ 2/[(x+h)2 + 16] - 2/(x2+16)] / h

Accepted Answer

Identify the complex fraction: the numerator is the difference of two fractions, and the denominator is a single variable $$h. $$The expression is: $$\frac{\frac{2}{(x+h)^2$ + 16} - \frac{2}{x^2$ + 16}}{h}. $$Focus on simplifying the numerator first, which is $$\frac{2}{(x+h)^2$ + 16} - \frac{2}{x^2$ + 16}. To $$combine these two fractions, find a common denominator, which is $$\$$left((x+h)^2$$ + 16\right) \$$left(x^2$$ + 16\right). $$Rewrite each fraction with the common denominator: $$\frac{2(x^2$ + 16)}{\$$left((x+h)^2$$ + 16\right) \$$left(x^2$$ + 16\right)} - \frac{2((x+h)^2$ + 16)}{\$$left((x+h)^2$$ + 16\right) \$$left(x^2$$ + 16\right)}. $$Combine the numerators over the common denominator: $$\frac{2(x^2$ + 16) - 2((x+h)^2$ + 16)}{\$$left((x+h)^2$$ + 16\right) \$$left(x^2$$ + 16\right)}. $$Now, rewrite the entire original expression as a single fraction by dividing this result by $$h$$, which is equivalent to multiplying by $$\frac{1}{h}. So $$the expression becomes: $$\frac{2(x^2$ + 16) - 2((x+h)^2$ + 16)}{h \cdot \$$left((x+h)^2$$ + 16\right) \$$left(x^2$$ + 16\right)}.$$

Simplify each complex fraction. [ 2/[(x+h)² + 16] - 2/(x²+16)] / h

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Key Concepts

Complex Fractions

Common Denominator

Difference Quotient and Simplification