91Ó°ÊÓ

The distributive property is a crucial concept in algebra. It allows us to multiply a single term by each term within a parenthesis.
This property states that for any numbers or variables \( a, b, \) and \( c \), the expression \( a(b + c) \) is equal to \( ab + ac \).
Let's use this property in our example:

Given expression: \[ 4a - b - 4(5a - 7b + 8c) \]
We distribute the \(-4\) to each term inside the parentheses:

• \(-4 \cdot 5a = -20a \)
• \(-4 \cdot -7b = 28b \) (Note: two negatives make a positive)
• \(-4 \cdot 8c = -32c \)

So after distribution:
\[ 4a - b - 20a + 28b - 32c \]

Combining like terms is another important step in simplifying algebraic expressions.
Like terms are terms that have the same variable raised to the same power.
We combine them by adding or subtracting their coefficients.

Let's identify and combine the like terms from our distributed expression:

• Variables with \( a \): \( 4a \) and \( -20a \)
• Variables with \( b \): \( -b \) and \( 28b \)
• Terms with \( c \): Only one term \( -32c \)

Now we combine these terms:

• \( 4a - 20a = -16a \)
• \( -b + 28b = 27b \)

Since there is only one term with \( c \), it remains \( -32c \).
Combining them all, we get: \[ -16a + 27b - 32c \]

Algebraic simplification involves reducing an expression to its simplest form by performing arithmetic operations and combining like terms.
This process makes the expression easier to understand and work with.

In our example, we have already applied the distributive property and combined like terms.
Let us put it all together to show the simplification process:

Original expression:
\[ 4a - b - 4(5a - 7b + 8c) \]
After distribution:
\[ 4a - b - 20a + 28b - 32c \]
After combining like terms:
\[ -16a + 27b - 32c \]

This final expression \( -16a + 27b - 32c \) is a simplified form.
Each term is unique, and no further combining of like terms is possible.

By understanding and applying these core concepts - the distributive property, combining like terms, and algebraic simplification - you can efficiently simplify complex algebraic expressions.