Question

complex fractions
simplify each expression.

1. \\(dfrac{dfrac{5}{4} + dfrac{m + 1}{4}}{5}\\)
2. \\(dfrac{dfrac{u}{4} + dfrac{4}{25}}{25}\\)
3. \\(dfrac{dfrac{4}{5} - dfrac{25}{x^2}}{16}\\)
4. \\(dfrac{dfrac{m + 5}{2}}{dfrac{m}{4} + dfrac{1}{m + 5}}\\)

Explanation:

Problem 1:

Step1: Simplify the numerator

The numerator is \(\frac{5}{4}+\frac{m + 1}{4}\). Since the denominators are the same, we add the numerators: \(\frac{5+(m + 1)}{4}=\frac{m+6}{4}\)

Step2: Divide by the denominator

Now we have \(\frac{\frac{m + 6}{4}}{5}\). Dividing by a number is the same as multiplying by its reciprocal, so \(\frac{m + 6}{4}\times\frac{1}{5}=\frac{m + 6}{20}\)

Step1: Simplify the numerator

The numerator is \(\frac{u}{4}+\frac{4}{25}\). Find a common denominator, which is \(100\). So \(\frac{u\times25}{4\times25}+\frac{4\times4}{25\times4}=\frac{25u+16}{100}\)

Step2: Divide by the denominator

Now we have \(\frac{\frac{25u + 16}{100}}{25}\). Multiply by the reciprocal of \(25\): \(\frac{25u+16}{100}\times\frac{1}{25}=\frac{25u + 16}{2500}\)

Step1: Simplify the numerator

The numerator is \(\frac{4}{5}-\frac{25}{x^{2}}\). Find a common denominator \(5x^{2}\). So \(\frac{4x^{2}}{5x^{2}}-\frac{125}{5x^{2}}=\frac{4x^{2}-125}{5x^{2}}\)

Step2: Divide by the denominator

Now we have \(\frac{\frac{4x^{2}-125}{5x^{2}}}{16}\). Multiply by the reciprocal of \(16\): \(\frac{4x^{2}-125}{5x^{2}}\times\frac{1}{16}=\frac{4x^{2}-125}{80x^{2}}\)

Answer:

\(\frac{m + 6}{20}\)

Problem 2: