Problem 62 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$8 y(12 a+b)$$

Step-by-Step Solution

Answer

Question: Use the distributive property to expand the following expression: $$8y(12a+b)$$. Answer: The expanded form of the given expression is $$96ay + 8yb$$.

Step 1: Identify the terms to distribute

The expression given is $$8y(12a+b)$$. Here, the term $$8y$$ needs to be distributed to both $$12a$$ and $$b$$ inside the parenthesis.

Step 2: Distribute the term

Using the distributive property, we will multiply $$8y$$ with both $$12a$$ and $$b$$: $$8y \cdot 12a + 8y \cdot b$$.

Step 3: Simplify the expression

Now, perform the multiplication: $$ (8y \cdot 12a) + (8y \cdot b) = (96ay) + (8yb)$$. So, the expanded form of the given expression is $$96ay + 8yb$$.

Key Concepts

Expanding Expressions

When we talk about expanding expressions in algebra, it's like opening up a folded paper to see all the contents at once. In mathematical terms, it involves simplifying an equation or expression so that there are no parentheses, and everything is laid out clearly. If we apply this to the given problem, we're taking the expression \(8y(12a+b)\) and 'expanding' it out.

In practice, we use the distributive property to do this, which basically says that you can distribute or multiply a single term to each term inside the parentheses and then add the results. It's kind of like giving out equal slices of a cake to each person at a party.

I like to imagine the term outside the parentheses, \(8y\) in our case, as a multiplier that 'touches' each term inside one by one. In the given exercise, \(8y\) multiplies with \(12a\) and then with \(b\), like shaking hands individually with two different people, to give us \(96ay + 8yb\).

Algebraic Expressions

Algebraic expressions are the phrases of the math world, combining numbers, variables (like \(x\), \(y\), and \(a\)), and operation signs into a compact form. Think of them as sentences that convey mathematical ideas without using an equals sign, which would make them equations.

Our expression \(8y(12a+b)\) is a chunk of algebraic terms ready for us to manage. The real beauty of these expressions is that they can represent real-life quantities and relationships, which can be manipulated through various properties like the distributive property we've used here.

While expanding is one way to handle algebraic expressions, we can also combine like terms, factor, or even solve for a variable if it's in equation form. The skill lies in recognizing what to do and when — almost like choosing the right tool for a job.

Multiplication in Algebra

Multiplication in algebra might just seem like it's about finding the product of numbers, but it's also the core technique we use when dealing with variables. When we multiply terms, especially when variables are involved, we're not just crunching numbers but combining units of measure, constants, and unknown quantities.

Expanding our example, when we multiply \(8y\) by \(12a\), we're not only multiplying the numbers together but also joining the variables. This multiplication is commutative, which means you can do it in any order; \(8y \times 12a\) is the same as \(12a \times 8y\). But remember, in algebra, we don’t write the multiplication sign. We just snug the numbers and letters close together to show they're being multiplied, ending up with \(96ay\) after combining \(8 \times 12\) and joining \(y\) and \(a\).

Similarly, \(8y \times b\) gives us \(8yb\). Again, the order doesn't matter, and you usually write it alphabetically, but here we've kept the order from the original problem to make our steps crystal clear.