Problem 62 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 2 x^{2} y^{4}\left(3 x^{2} y+4 x y+3 y\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(2x^{2}y^{4}(3x^{2}y+4xy+3y)\) Answer: \((6x^{4}+8x^{3}+6x^{2})y^{5}\)

Step 1: Identify the expression

We are given the expression: $$ 2x^{2}y^{4}\left(3x^{2}y+4xy+3y\right) $$ Our goal is to simplify this expression. Step 2: Distribute the polynomial

Step 2: Distribute the polynomial

Using the distributive property, we distribute the \(2x^{2}y^{4}\) term to each term within the parentheses: $$ 2x^{2}y^{4} \times 3x^{2}y + 2x^{2}y^{4} \times 4xy + 2x^{2}y^{4} \times 3y $$ Step 3: Apply the laws of exponents

Step 3: Apply the laws of exponents

Recall that when we multiply terms with the same base, we add their exponents. Apply this rule to the terms in the expression: $$ 6x^{2+2}y^{4+1} + 8x^{2+1}y^{4+1} + 6x^{2}y^{4+1} $$ Step 4: Simplify the exponents

Step 4: Simplify the exponents

Now, we simplify the exponents in each term of the expression: $$ 6x^{4}y^{5} + 8x^{3}y^{5} + 6x^{2}y^{5} $$ Step 5: Combine like terms

Step 5: Combine like terms

Finally, combine the terms with the same variables and exponents: $$ 6x^{4}y^{5} + 8x^{3}y^{5} + 6x^{2}y^{5} = (6x^{4}+8x^{3}+6x^{2})y^{5} $$ The simplified algebraic expression is: $$ (6x^{4}+8x^{3}+6x^{2})y^{5} $$

Key Concepts

Distributive Property

Understanding the distributive property is fundamental in simplifying algebraic expressions. Picture distribute, the keyword, as handing out pieces of a whole. Mathematically, it allows you to multiply a single term by each term within a parenthesis.

For example, in the expression 2x^{2}y^{4}(3x^{2}y+4xy+3y), you multiply 2x^{2}y^{4} with each term inside the parentheses: 3x^{2}y, 4xy, and 3y. The distributive property makes this step systematic and helps to expand the expression which later can be simplified further.

Application of the distributive property ensures that every term is considered and that no part of the expression is left unmultiplied, thereby reducing the chances of error and laying the groundwork for further simplification steps.

Laws of Exponents

The laws of exponents govern how to handle powers when they're being multiplied, divided, or raised to other powers. When multiplying terms that have the same base, the rule is to add their exponents. It's like stacking levels of a building; you go up one level at a time.

In our example, when we distribute 2x^{2}y^{4} across the parentheses and multiply x^{2} with x^{2}, we follow the law and add the exponents: 2 + 2 to get x^{4}. This applies to the y terms as well. The laws of exponents simplify the process, turning a potentially complex multiplication into a straightforward addition of exponents.

Combine Like Terms

Combining like terms is like sorting fruits into baskets; you group the same types together to see what you have more clearly. In algebra, like terms are terms that have the exact same variables raised to the same power.

In the expression 6x^{4}y^{5} + 8x^{3}y^{5} + 6x^{2}y^{5}, each term has an exponent of 5 for y, but different powers of x. This suggests that they cannot be combined further. Combining like terms makes the expression cleaner and more manageable. It’s a crucial step for achieving the most simplified version of an expression.

Polynomial Operations

What happens when we deal with more than monomials? Polynomial operations come into play. A polynomial can be thought of as a family of terms united under addition or subtraction signs. Operations on polynomials include addition, subtraction, multiplication, and, in some cases, division.

The final simplified form (6x^{4}+8x^{3}+6x^{2})y^{5} of our initial expression is the result of carrying out polynomial operations efficiently. By knowing how to handle each term with care, respecting the hierarchy of algebraic operations, and appropriately applying the distributive property, laws of exponents, and combining like terms, we achieve mastery over polynomials. This mastery leads to precise and simplified solutions in algebra.