Problem 39
Question
Simplify \((a+5 a) \times 2 a-3 a\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(12a^2 - 3a\).
1Step 1: Combine Like Terms Inside the Parentheses
Look at the expression inside the parentheses: \((a + 5a)\). Here, both terms have \(a\) as the variable. Combine the coefficients: \(1a + 5a = 6a\). So the expression simplifies to \(6a\).
2Step 2: Distribute the Multiplication
Now, we have \((6a) \times 2a - 3a\). Distribute the multiplication over \(6a\); calculate \((6a) \times 2a = 12a^2\). This simplifies the expression further: \(12a^2 - 3a\).
3Step 3: Identify and Simplify Remaining Terms
In the expression \(12a^2 - 3a\), no further like terms can be combined since \(12a^2\) and \(-3a\) have different exponents. Thus, the simplified expression is \(12a^2 - 3a\).
Key Concepts
Combining Like TermsDistributive PropertyExpression Simplification
Combining Like Terms
When working with algebraic expressions, one fundamental skill to master is **combining like terms**. Like terms are those terms in an expression that have the same variable raised to the same power.
For instance, in the expression within the parentheses \(a + 5a\), both terms consist of the variable \(a\) raised to the same power.
To combine these like terms, you only need to add or subtract their coefficients (the numbers in front of the variables).
Here, we add the coefficients: \(1a + 5a = 6a\).
For instance, in the expression within the parentheses \(a + 5a\), both terms consist of the variable \(a\) raised to the same power.
To combine these like terms, you only need to add or subtract their coefficients (the numbers in front of the variables).
Here, we add the coefficients: \(1a + 5a = 6a\).
- Always ensure the variables and their exponents match.
- Different variables or exponents mean the terms are not like terms and cannot be combined.
- Think of combining like terms as gathering similar items together to simplify your expression.
Distributive Property
The **distributive property** is a key concept in simplifying algebraic expressions, especially when dealing with multiplication over addition or subtraction.
This property allows you to distribute a multiplier across terms within parentheses to simplify the expression. For example, consider the expression \( (6a)\times 2a - 3a \).The distributive property enables us to break this down by multiplying \(6a \) by each term in its proximity:
This property allows you to distribute a multiplier across terms within parentheses to simplify the expression. For example, consider the expression \( (6a)\times 2a - 3a \).The distributive property enables us to break this down by multiplying \(6a \) by each term in its proximity:
- Multiply: \( (6a)\times 2a = 12a^2 \).
- Multiply each term connected by a plus or minus sign separately.
- Maintain operations (addition or subtraction) as you simplify.
Expression Simplification
After using the distributive property and combining like terms, the next task is **expression simplification**.
This involves looking at your expression and seeing if any further simplification can be done.
The expression \(12a^2 - 3a\) cannot be simplified any further because:
Always check if all algebraic operations have been resolved, ensuring no further factors can be distributed or combined in the expression.
This involves looking at your expression and seeing if any further simplification can be done.
The expression \(12a^2 - 3a\) cannot be simplified any further because:
- \(12a^2\) and \(-3a\) have different exponents, making them unlike terms.
- Without like terms, no further combination or reduction can occur.
Always check if all algebraic operations have been resolved, ensuring no further factors can be distributed or combined in the expression.