Problem 39 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(10 \cdot[8+2 \cdot(6+7)]\)

Step-by-Step Solution

Answer

The expression evaluates to 340.

Step 1: Precedence: Parentheses First

Look inside the innermost parentheses and solve. Since \((6+7)\) is inside the deepest parentheses, calculate that first. So, \(6+7=13\).

Step 2: Apply Multiplication within Parentheses

Now replace the expression with the previously solved value. The expression becomes \(10 imes [8 + 2 imes 13]\).Now calculate \(2 \times 13\).The result is 26, so the expression is \(10 \times [8 + 26]\).

Step 3: Solve Inside Brackets

Calculate the expression inside the brackets \((8+26)\). The result is 34. So the expression is now \(10 \times 34\).

Step 4: Final Multiplication

Now, multiply the remaining numbers outside the brackets.Calculate \(10 \times 34\). This results in 340.

Key Concepts

Parentheses

When working with arithmetic expressions, parentheses are used to alter the standard order of operations. Parentheses indicate which operations should be performed first, overriding the typical sequence of operations known as PEMDAS/BODMAS (Parentheses/Brackets, Orders/Exponents, Multiplication, Division, Addition, Subtraction).
Within a set of parentheses, follow the same PEMDAS/BODMAS rules. Solve any inner parentheses or brackets first, moving outward until you solve within each enclosing parenthesis.

• Start with the innermost parentheses and work outward.
• Perform all operations within the parentheses before continuing with the rest of the expression.

In our exercise, we find \(6+7\) inside the parentheses, which we solve first. By understanding and correctly applying the concept of parentheses, you can analyze and simplify arithmetic expressions in a logical and orderly fashion.

Multiplication

Multiplication is a key operation in arithmetic and often appears explicitly or implicitly in expressions. In the context of order of operations, multiplication follows parentheses. This means that after evaluating the contents of any parentheses, multiplication, along with division, is the next priority in our sequence of calculations.
Multiplication can be thought of as repeated addition. For instance, multiplying 2 by 13 (as in our exercise step) means adding 13 twice. The expression simplifies step by step to allow us to focus on the multiplication component next.

• First, complete any arithmetic inside parentheses.
• Then, handle multiplication and division from left to right as it appears in the expression.

In our specific problem, once we ascertain the value within the parentheses as \(13\), we then perform the multiplication of \(2 imes 13\) to get 26. Understanding multiplication's role is crucial in correctly simplifying expressions.

Arithmetic Expressions

Arithmetic expressions combine numbers and operation symbols to form a statement. They can include operations such as addition, subtraction, multiplication, and division. In our problem, we dealt with a more complex nested expression, requiring careful application of the order of operations.
An arithmetic expression like \(10 \cdot [8+2 \cdot (6+7)]\) incorporates parentheses to define the sequence of operations clearly. Brackets function similarly to parentheses, guiding the steps of solving arithmetic expressions carefully. This helps ensure accuracy.

• Follow the sequence: Solve any parts of the expression enclosed in parentheses or brackets first.
• Employ multiplication and division in the sequence they appear next, working left to right.
• Lastly, address addition and subtraction.

Breaking down the problem into manageable steps simplifies the process, allowing us to clearly see, step-by-step, how complex expressions simplify to a single numerical value, such as the final result 340 in our example.