Question: 7. Use Order of Operations to simplify. Show work. a) \(\left(\frac{2}{7} +…

7. Use Order of Operations to simplify. Show work.

a) \(\left(\frac{2}{7} + \frac{3}{14}\right) \left(\frac{2}{7} - \frac{3}{14}\right)\)

Solution

To simplify the expression \(\left( \frac{2}{7} + \frac{3}{14} \right)\left( \frac{2}{7} - \frac{3}{14} \right)\), follow these steps: First, find a common denominator for the fractions \(\frac{2}{7}\) and \(\frac{3}{14}\). The least common denominator of 7 and 14 is 14. Rewrite \(\frac{2}{7}\) as \(\frac{4}{14}\). Now the expression becomes: \[ \left( \frac{4}{14} + \frac{3}{14} \right)\left( \frac{4}{14} - \frac{3}{14} \right) \] Simplify each part inside the parentheses: \[ = \left( \frac{4 + 3}{14} \right)\left( \frac{4 - 3}{14} \right) \] Simplify further: \[ = \left( \frac{7}{14} \right)\left( \frac{1}{14} \right) \] Reduce \(\frac{7}{14}\) to \(\frac{1}{2}\): \[ = \left( \frac{1}{2} \right)\left( \frac{1}{14} \right) \] Multiply the fractions: \[ = \frac{1 \times 1}{2 \times 14} = \frac{1}{28} \] The simplified form of the expression is \(\frac{1}{28}\).