Problem 55 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 2\left\\{5\left[6\left(b+2 a+c^{2}\right)\right]\right\\} $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: $$ 2\left\{5\left[6\left(b+2a+c^{2}\right)\right]\right\} $$ Answer: $$ 120a + 60b + 60c^{2} $$

Step 1: Distribute the constants inside the brackets

First, distribute the innermost constant 6 inside the parentheses (b + 2a + c^2): $$ 2\left\{5\left[6b+12a+6c^{2}\right]\right\} $$

Step 2: Distribute the constants inside the braces

Next, distribute the constant 5 inside the brackets [6b + 12a + 6c^2]: $$ 2\left\{30b+60a+30c^{2}\right\} $$

Step 3: Distribute the constant outside the braces

Finally, distribute the constant 2 inside the braces {30b + 60a + 30c^2}: $$ 60b+120a+60c^{2} $$

Step 4: Write the simplified expression

Now that we've distributed all the constants and combined like terms, we can write the simplified algebraic expression: $$ 120a + 60b + 60c^{2} $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that enables us to multiply a single term by each term inside a parenthesis or bracket. Essentially, it helps us simplify expressions where multiplication is distributed over addition or subtraction.

Consider an expression like \( a(b + c) \). The distributive property tells us that you multiply \( a \) by both \( b \) and \( c \), resulting in \( ab + ac \). This is precisely what we did in Step 1 of our exercise, where the number 6 was distributed over the terms inside the parentheses which contained \( b \), \( 2a \), and \( c^2 \).

Understanding how to correctly apply the distributive property can make complex expressions much simpler to handle. It's analogous to sharing an equal amount of something among a few friends – everyone gets their fair portion. In algebra, every term within the brackets gets its 'fair share' of the outside number.

Combining Like Terms

Once we have used the distributive property to expand an algebraic expression, we often need to look out for like terms to further simplify the expression. Like terms are terms that have the same variable parts raised to the same powers, which means their coefficients can be combined through addition or subtraction.

For example, \( 3a + 5a \) can be simplified by combining the coefficients of \( a \) to get \( 8a \). However, terms like \( 2a \) and \( 3b \) are not like terms because they have different variable parts.

In the final steps of our exercise, we applied this concept after expanding the entire expression. The process we followed didn’t create additional like terms, but in cases where it does, combining them is a key step to reach the simplest form of the expression. This could be thought of as gathering all similar fruits into their respective baskets to see what you have at a glance.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \) or \( y \)), and operators (like addition and multiplication). Expressions are the building blocks of algebra and are used to represent relationships between different quantities.

In our exercise, \( 2\big\{5\big[6(b+2a+c^2)\big]\big\) is an example of an algebraic expression. It includes numerical coefficients (2, 5, 6), variables (\( a, b, c \) ), and exponents (as seen in \( c^2 \) which stands for \( c \times c \) ). There are no equal signs in an expression, so they don’t state an equality like an equation would, but they can be simplified or manipulated algebraically.

Learning to work with algebraic expressions is like learning a new language; each part has a role, and when put together correctly, they convey meaningful information which can be used to solve problems.