Problem 61 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$13+(7-2)(5-1)$$

Step-by-Step Solution

Answer

The simplified expression is 33.

Step 1: Solve the parentheses

First, solve the expressions inside the parentheses. The first set of parentheses is \(7-2\) and the second set is \(5-1\). Calculate each separately.\(7-2=5\) and \(5-1=4\).

Step 2: Multiply the results

Now, multiply the results obtained from the parentheses. So, we have \((5)(4)\). Multiply to get \(5 \times 4 = 20\).

Step 3: Add to the remaining expression

Finally, take the result from the multiplication in Step 2 and add it to the remaining number outside the parentheses, which is 13. This is \(13 + 20\). Add them together to get \(13 + 20 = 33\).

Key Concepts

Simplification Process

Numerical expression simplification is a fundamental math skill. There are usually several steps to simplify an expression. 1. **Evaluate expressions inside parentheses**: This is always the first priority in mathematics. 2. **Perform multiplications or divisions**: These operations typically follow the solving of parentheses. 3. **Finish with additions or subtractions**: Once parentheses are solved and other operations are complete, additions or subtractions tie everything together. This order of operations is essential because it gives a universally accepted way of handling mathematical expressions, ensuring consistency in solving numerical expressions.

Parentheses Solving

Parentheses are crucial in expressions as they indicate specific computations that must happen first. To deal with them:- Identify all pairs of parentheses.- Solve any operations inside them before moving on to other parts of the expression.- Think of parentheses as "high-priority zones"—anything within them takes precedence over other operations.For example, in the expression \(13 + (7-2)(5-1)\):- Solve \(7-2\) to get \(5\) and \(5-1\) to get \(4\) before anything else happens. By dealing with what’s in parentheses early on, you simplify the problem considerably.

Step-by-Step Solution

Breaking down problems in a step-by-step manner helps students track their thought processes. As seen in our original solution:- **Step 1**: Solve expressions within parentheses, like \(7-2\) and \(5-1\).- **Step 2**: Multiply the results from the parentheses; i.e., \((5)(4)\).- **Step 3**: Add those results with the remaining numerical components, here \(13 + 20\).This sequence clearly demonstrates how to tackle each part of the expression methodically. Monitoring each individual step can serve as a confirmation of accuracy for the student.