Problem 77 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 8 a[2 a-4 a b+9(a-5-a b)] $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: $$ 8a[2a - 4ab + 9(a - 5 - ab)] $$ Answer: $$ 88a^2 - 104a^2b - 360a $$

Step 1: Distribute 8a inside the expression

We will distribute 8a to each term inside the brackets. $$ 8a[2a - 4ab + 9(a - 5 - ab)] = 8a(2a) - 8a(4ab) + 8a[9(a - 5 - ab)] $$

Step 2: Distribute 9 inside the innermost brackets

Distribute the 9 to each term inside the innermost brackets. $$ 8a(2a) - 8a(4ab) + 8a[9(a - 5 - ab)] = 8a(2a) - 8a(4ab) + 8a(9a - 45 - 9ab) $$

Step 3: Simplify each term

Now, we will multiply each term as required. $$ 8a(2a) - 8a(4ab) + 8a(9a - 45 - 9ab) = 16a^2 - 32a^2b + 72a^2 - 360a - 72a^2b $$

Step 4: Combine like terms

Combine the like terms to simplify the expression: $$ 16a^2 + 72a^2 - 32a^2b - 72a^2b - 360a = 88a^2 - 104a^2b - 360a $$ The simplified expression is: $$ 88a^2 - 104a^2b - 360a $$

Key Concepts

Algebraic Distribution

Algebraic distribution is like sharing or distributing a value across other values within brackets in an expression. It happens when you multiply a term outside the bracket with every term inside the bracket. Let's understand this step-by-step:

In the expression given:

• 8a \[2a - 4ab + 9(a - 5 - ab)\]

Start by multiplying the term outside the bracket, 8a, to each term inside the first bracket. This means:

• 8a \times 2a = 16a^2
• 8a \times (-4ab) = -32a^2b
• The next part is to handle the expression inside the second set of brackets.

Inside the nested bracket 9(a - 5 - ab), apply the distribution of 9, which gives:

• 9 \times a = 9a
• 9 \times (-5) = -45
• 9 \times (-ab) = -9ab

Combine these results back into the original expression at each step to further continue simplifying.

Combining Like Terms

Once you've applied algebraic distribution and simplified each multiplication step, the next important task is to spot terms that can be joined because they are similar, also known as combining like terms. Like terms refer to elements that have the same variables raised to the same power:

Consider:

• 16a^2 and 72a^2, both consist of the variable a^2
• -32a^2b and -72a^2b, which contain the terms a^2b
• The standalone linear term -360a

To effectively combine them:

• Add coefficients of terms with the same powers resulting in 88a^2 when you sum 16a^2 and 72a^2
• Doing similarly for -32a^2b and -72a^2b gives -104a^2b
• The term -360a stays as it is because it doesn't combine with any other term.

Combining like terms simplifies the expression significantly by reducing the number of terms and focusing only on critical, non-redundant elements.

Polynomial Simplification

Polynomial simplification involves reducing an expression containing multiple terms into a simpler or more compact form without changing its value. It's like making a long sentence concise but retaining the same meaning. Here's how you approach it with our exercise:

After distributing and combining like terms, the polynomial expression becomes smaller:

• Instead of working with each original long bracket expression separately, put it all together for clarity.
• The simplified form from the steps becomes 88a^2 - 104a^2b - 360a.
• Breaking it down, you have two quadratic terms and one linear term.

By doing this, you get a neat version of the original complex expression. Simplifying polynomials gives a clearer understanding of relationships between terms and helps in further calculations if needed.