Simplifying Algebraic Expressions Step-by-Step

We need to simplify the given algebraic expression: \(7p - [3q - \{8p - (4q - 10p)\}]\).

To simplify this expression, we will follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS, which means we simplify the parts inside the brackets, then the curly braces, and finally the square brackets.

Step 1: Simplify the innermost parentheses

The innermost part is \((4q - 10p)\). There are no like terms or operations to perform inside, so we remove the parentheses. The expression becomes:

\(7p - [3q - \{8p - (4q - 10p)\}] = 7p - [3q - \{8p - 4q + 10p\}]\)

Note that the negative sign before the parentheses is distributed to both terms inside.

Step 2: Simplify the expression inside the curly braces

The expression inside the curly braces is \(\{8p - 4q + 10p\}\). We can combine the like terms involving \(p\): \(8p + 10p = 18p\).

So, the expression inside the curly braces simplifies to \(\{18p - 4q\}\).

The main expression now looks like:

\(7p - [3q - \{18p - 4q\}]\)

Step 3: Simplify the expression inside the square brackets

The expression inside the square brackets is \([3q - \{18p - 4q\}]\). We need to remove the curly braces, remembering to distribute the negative sign before them.

\([3q - (18p - 4q)] = [3q - 18p + 4q]\)

Now, combine the like terms inside the square brackets, which are the terms involving \(q\): \(3q + 4q = 7q\).

So, the expression inside the square brackets simplifies to \([7q - 18p]\).

The main expression is now:

\(7p - [7q - 18p]\)

Step 4: Complete the simplification

Finally, we remove the square brackets. Again, distribute the negative sign before the square brackets to each term inside.

\(7p - (7q - 18p) = 7p - 7q + 18p\)

Combine the like terms, which are the terms involving \(p\): \(7p + 18p = 25p\).

The simplified expression is \(25p - 7q\).

Let's summarize the steps:

1. Original expression: \(7p - [3q - \{8p - (4q - 10p)\}]\)
2. Remove parentheses: \(7p - [3q - \{8p - 4q + 10p\}]\)
3. Combine like terms in curly braces: \(7p - [3q - \{18p - 4q\}]\)
4. Remove curly braces: \(7p - [3q - 18p + 4q]\)
5. Combine like terms in square brackets: \(7p - [7q - 18p]\)
6. Remove square brackets: \(7p - 7q + 18p\)
7. Combine like terms: \(25p - 7q\)

The simplified form of the expression \(7p - [3q - \{8p - (4q - 10p)\}]\) is \(25p - 7q\).

Revision Table: Key Concepts in Algebraic Simplification

Additional Information: Handling Nested Brackets

When simplifying expressions with nested brackets like parentheses \(()\), curly braces \(\{\}\), and square brackets \(\{\}\), it is standard practice to work from the innermost set of grouping symbols outwards. This ensures that the operations within each level are correctly evaluated before they are affected by operations or signs outside.

• Start with the operations or simplifications inside the innermost parentheses \(()\).
• Once the parentheses are simplified or removed (by distributing any sign or factor outside), move to the curly braces \(\{\}\).
• After the curly braces are simplified or removed, proceed to the square brackets \(\{\}\).
• Finally, perform any remaining operations outside the brackets.

Always be careful when distributing a negative sign (\(-\)) or a number outside a set of brackets. The sign or number must be multiplied by every term inside those brackets.