Problem 17 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$2 \frac{3}{8}+\frac{1}{2}\left(\frac{1}{3}+\frac{5}{3}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is 6 \( \frac{3}{8} \).

Step 1: Solve the Inner Parentheses

First, evaluate the expression inside the parentheses: \( \frac{1}{3} + \frac{5}{3} \). Since the denominators are the same, simply add the numerators: \( \frac{1+5}{3} = \frac{6}{3} \). Thus, it simplifies to 2.

Step 2: Exponentiation

Next, apply the exponent to the result from Step 1: \( 2^3 \). Calculate this as follows: \( 2 \times 2 \times 2 = 8 \). So \( \left( \frac{1}{3} + \frac{5}{3} \right)^3 = 8 \).

Step 3: Multiply by the Fraction Outside the Parentheses

Now, multiply \( \frac{1}{2} \) by the result from Step 2: \( \frac{1}{2} \times 8 = 4 \).

Step 4: Final Addition

Finally, add \( 2 \frac{3}{8} \) to 4. Convert \( 2 \frac{3}{8} \) to an improper fraction: \( \frac{19}{8} \). Then, find a common denominator to add it to 4. Convert 4 into a fraction with denominator 8: \( \frac{32}{8} \). Add the fractions: \( \frac{19}{8} + \frac{32}{8} = \frac{51}{8} \). Convert \( \frac{51}{8} \) back into a mixed number: 6 \( \frac{3}{8} \).

Key Concepts

Parentheses

When faced with complex mathematical expressions, parentheses help us determine where to start first. Mathematicians use the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to decide the sequence for solving pieces of an expression.

In the equation given, solving what's inside the parentheses comes first:

• Using parentheses groups terms together and shows that those terms should be calculated as a unit.
• Here, within the parentheses, you have \( \frac{1}{3} + \frac{5}{3} \).
• The denominators are the same, so just add the numerators (1 + 5) to get \( \frac{6}{3} \), which simplifies to 2.

Doing calculations inside parentheses before moving on ensures precision and accuracy.

Exponentiation

Once you solve inside the parentheses, you often find an exponent waiting next. Exponentiation means multiplying a number by itself a specified number of times. For example,

• The expression from the problem, \( (2)^3 \), translates to multiplying 2 three times: 2 x 2 x 2.
• Following the steps, this yields 8 as the result of exponentiation.

This step is crucial right after solving the parentheses because it sets up the value that influences all following calculations. Approaching exponentiation with care can prevent errors in this vital step.

Fractions

Fractions might seem challenging, but they follow simple arithmetic rules. Adding, subtracting, or multiplying fractions calls for some attention.

• For addition, like in our example, a common denominator is necessary. With \( \frac{1}{3} + \frac{5}{3} \), we add numerators directly since the denominator remains the same.
• Multiplication, as shown when multiplying \( \frac{1}{2} \times 8 \), involves multiplying the numerator by the number, which results in 4.

Always break down fractions step-by-step. This approach ensures active management of numerators and denominators while simplifying complexity with precision.

Mixed Numbers

Mixed numbers combine whole numbers and fractions. They appear in many practical scenarios and exercises. Knowing how to convert and manage them enhances mathematical fluency.

• In the example, to add \( 2 \frac{3}{8} \) and 4, first turn the mixed number into an improper fraction. The \( 2 \frac{3}{8} \) becomes \( \frac{19}{8} \).
• Similarly, change 4 to an equivalent fraction format: \( \frac{32}{8} \).
• Then, add these fractions, \( \frac{19}{8} + \frac{32}{8} = \frac{51}{8} \), and convert back to 6 \( \frac{3}{8} \).

Simplifying mixed numbers by moving between improper fractions and back not only reveals their utility but also builds flexibility for tackling a wide range of math problems.