Evaluate the following expression

$\frac{13}{24}$

Explanation

Strategy We will scan the expression to determine what operations need to be performed. Then we will perform those operations, one-at-a-time, following the order of operations rule.

WHY If we don’t follow the correct order of operations, the expression can have more than one value.

Solution
Although the expression contains parentheses, there are no calculations to perform within them. We will begin by evaluating all exponential expressions. We will write the steps of the solution in horizontal form.

• Evaluate: \(\left(-\frac12 \right)^3 = \left( -\frac12 \right) \left( -\frac12 \right) \left( -\frac12 \right) = -\frac18\)
  • \(\frac34 + \frac53 \left( -\frac12 \right)^3 = \frac34 + \frac53 \left(-\frac18 \right)\)
• Multiply: \(\frac53 \left(-\frac18\right) = -\frac{5 \cdot 1}{3 \cdot 8} = -\frac{5}{24}\).
  • = \(\frac34 + \left( -\frac{5}{24}\right)\)
• Prepare to add the fractions: Their LCD is 24.  To build the first fraction so that its denominator is 24, multiply it by a form of 1.
  • = \(\frac34 \cdot \frac66 + \left( -\frac{5}{24}\right)\)
• Multiply the numerators: 3 • 6 = 18.  Multiply the denominators: 4 • 6 = 24
  • = \(\frac{18}{24} + \left( -\frac{5}{24} \right)\)
• Add the numerators: 18 + (-5) = 13.  Write the sum over the common denominator 24.
  • = \(\frac{13}{24}\)

If an expression contains grouping symbols, we perform the operations within the grouping symbols first.