Problem 29 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ 2 x^{2}\left(6 y^{3}\right)\left(2 x^{2} y\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(24x^4y^4\).

Step 1: Identify and reorganize the expression

The given expression is \(2x^2(6y^3)(2x^2y)\). Identify all the terms: a coefficient of 2, \(x^2\), \(6y^3\), \(2x^2\), and \(y\). We will reorganize these terms by grouping similar bases together. Rewrite the expression as \(2 \cdot 6 \cdot 2 \cdot x^2 \cdot x^2 \cdot y^3 \cdot y\).

Step 2: Simplify the coefficients

Multiply the numerical coefficients together. Calculate \(2 \times 6 \times 2 = 24\). Now our expression looks like \(24 \cdot x^2 \cdot x^2 \cdot y^3 \cdot y\).

Step 3: Simplify the powers of \(x\)

Combine the powers of \(x\) using the rule \(x^a \cdot x^b = x^{a+b}\). Thus, \(x^2 \cdot x^2 = x^{2+2} = x^4\). After this step, the expression becomes \(24 \cdot x^4 \cdot y^3 \cdot y\).

Step 4: Simplify the powers of \(y\)

Combine the powers of \(y\) using the similar exponential rule: \(y^3 \cdot y = y^{3+1} = y^4\). Finally, the expression is \(24x^4y^4\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For instance, when we see something like \(x^3\), it means \(x\) multiplied by itself three times: \(x \cdot x \cdot x\). Exponents simplify expressions, allowing us to write long products as a single term.
In algebra, each exponent tells us how many times the base (in this case, \(x\)) is used as a factor. Understanding how to work with exponents is crucial because they appear frequently in equations and formulas. They help reduce long multiplications to more manageable forms.
When simplifying expressions with exponents, remember that the bases must be the same to combine them. So, \(x^a \cdot x^b\) becomes \(x^{a+b}\). This property makes it easier to handle expressions efficiently. Keep this rule in mind as we dig deeper into these concepts!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). These expressions don't have an equals sign, unlike algebraic equations.
An example of an algebraic expression is \(3x + 2y - 5\). Notice how it consists of numbers (3, 2, and 5), variables (\(x\) and \(y\)), and operators (+ and -). In simplifying such expressions, knowing how to manipulate these components is essential.

• **Variables**: symbols like \(x\) or \(y\) that represent unknown numbers.
• **Constants**: numbers like 3 or 5 that have a fixed value.
• **Coefficients**: numbers in front of the variables that indicate how many times to multiply the variable (e.g., 3 in 3x).

Understanding these elements will make it easier to approach any algebraic problem. Remember, when simplifying, you aim to gather like terms together and perform any possible arithmetic operations to shorten or understand the expression clearly.

Product of Powers Rule

The "Product of Powers Rule" is an essential tool in simplifying expressions with exponents. It states that when you multiply terms that have the same base, you can add their exponents. For example, given \(x^a \cdot x^b\), this can be simplified to \(x^{a+b}\).
This rule is powerful because it lets us simplify expressions quickly without resorting to lengthy multiplication. It's crucial when dealing with complicated expressions, as it helps to break them into more straightforward components.
In the context of our exercise, we applied this rule twice:

• For the \(x\) terms: \(x^2 \cdot x^2 = x^{2+2} = x^4\). This consolidates the powers of the same base.
• For the \(y\) terms: \(y^3 \cdot y = y^{3+1} = y^4\). Again, we combined similar bases by adding their exponents.

Having a firm grasp of the "Product of Powers Rule" is crucial for anyone dealing with algebra, as it provides a quick method to simplify expressions and solve problems more efficiently.