Problem 114 - Elementary and Intermediate Algebra

Question

a. \(8+3[-2-(6+1)]\) b. \((8+3)[-2-(6+1)]\)

Step-by-Step Solution

Answer

a. -19 b. -99

Step 1a: Simplify Inside the Parentheses

For the expression in part (a), start by simplifying inside the innermost parentheses: \( (6+1) \). This gives \( (6+1) = 7 \). The expression now becomes \( 8 + 3[-2 - 7] \).

Step 2a: Simplify Inside the Brackets

Now, simplify the expression inside the brackets: \( -2 - 7 \). This results in \( -2 - 7 = -9 \). The expression is now \( 8 + 3[-9] \).

Step 3a: Perform Multiplication

Multiply 3 by the result from the brackets: \( 3 imes (-9) = -27 \). The expression is now \( 8 - 27 \).

Step 4a: Add or Subtract the Terms

Finally, compute \( 8 - 27 \), which equals \( -19 \). Thus, the value of the expression in part (a) is \( -19 \).

Step 1b: Simplify Inside the Parentheses

For the expression in part (b), simplify inside the innermost parentheses: \( (6+1) \) which gives \( (6+1) = 7 \). The expression now becomes \( (8+3)[-2 - 7] \).

Step 2b: Simplify Outside the Inner Parentheses

Simplify the expression outside the inner parentheses first: \( 8+3 \), which gives \( 11 \). The expression is now \( 11[-9] \).

Step 3b: Perform Multiplication

Now, multiply \( 11 \) by the number inside the brackets: \( 11 imes (-9) = -99 \). Thus, the value of the expression in part (b) is \( -99 \).

Key Concepts

Expressions in Algebra

Understanding expressions in algebra is like learning a new language. It's all about combining numbers and letters using different operations to express a mathematical idea. An algebraic expression can include numbers, variables (which stand for unknown values), and operators like plus, minus, multiply, and divide. For example, in the expression given in the exercise, we see numbers such as 8 and 3, and operations like addition and multiplication.
When working with algebraic expressions, always pay attention to the sequence of calculations. This sequence is crucial because it changes the outcome of the expression. Think of it as solving a puzzle step by step, each piece of information leading up to your final answer.
One must also familiarize themselves with terms like 'brackets', 'parentheses', and how to evaluate expressions, as these are fundamental in understanding algebra. The exercise above demonstrates how to evaluate an expression correctly by carefully following each operation according to the order of operations.

Simplifying Parentheses

Parentheses are used in algebra to indicate which operations should be done first. In the given exercise, parentheses play a crucial role in guiding the order of calculations. Simplifying what’s inside the parentheses first ensures that the operations are performed correctly.
Consider the expression part (a) from the exercise: \( 8+3[-2-(6+1)] \). The first step is simplifying the innermost parentheses, \( (6+1) \), to get 7. This simplification reduces complexity, allowing more straightforward operations afterward.
Remember these tips when simplifying parentheses:

• Always start with the innermost expression.
• Perform operations inside out, gradually opening up the expression.
• Each simplification step leads to a simpler expression that's easier to evaluate.

Using these steps correctly navigates you through even the most complicated-looking expressions.

Multiplication and Addition in Algebra

In algebra, multiplication and addition are core operations used to simplify expressions. They are guided by the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
In both parts (a and b) of the exercise, after simplifying inside parentheses, the next steps involve dealing with multiplication and addition.
Multiplication, like \( 3 \times (-9) \), is executed before any addition or subtraction operations unless guided otherwise by parentheses. This is because multiplication falls earlier in the order of operations.
The final steps bring addition or subtraction into play to complete the expression. In part (a), after calculating \( 3 \times (-9) \) to get \(-27\), you move to calculating \( 8 - 27 \), giving the final answer \(-19\). Following these rules diligently ensures accuracy in your outcomes.