Problem 45 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \left(12 x y^{2}\right)\left(-2 x^{3} y^{2}\right) $$

Step-by-Step Solution

Answer

The simplified form of \( (12x y^2) (-2x^3 y^2) \) is \(-24x^4 y^4\).

Step 1: Recognize The Structure

The expression to be simplified is a product of two terms, namely \(12xy^2\) and \(-2x^3y^2\). There are numerical factors and variable factors in each term.

Step 2: Apply Laws of Exponents

Whenever you multiply terms with the same base, add the exponents. In our case, the base is \(x\), and thus \(x^1\) from the first term and \(x^3\) from the second term are combined to get \(x^4\). Similarly, \(y^2\) from the first term and \(y^2\) from the second term are combined to get \(y^4\).

Step 3: Multiply Numerical Factors

Multiply the numerical factors \(12\) and \(-2\) to get \(-24\). The given expression is, therefore, simplified to \(-24x^4y^4\).

Key Concepts

Laws of Exponents

When working with algebraic expressions, exponent rules are essential for simplification. These rules apply when you're multiplying, dividing, or raising powers to powers.

Understanding the laws of exponents can make complex problems more manageable. Here are the key rules you'll often use:

• Product of Powers: When multiplying two terms with the same base, you add the exponents. For example, \(x^a * x^b = x^{a+b}\).
• Power of a Power: When raising an exponent to another power, you multiply the exponents. For instance, \( (x^a)^b = x^{a*b}\).
• Quotient of Powers: When dividing terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This looks like \(\frac{x^a}{x^b} = x^{a-b}\).
• Zero Exponent: Any base with an exponent of zero equals one. \((x^0 = 1\))
• Negative Exponent: A negative exponent indicates reciprocal. \(x^{-a} = \frac{1}{x^a}\).

When the expression \(12xy^2\) is multiplied by \( -2x^3y^2 \), according to the Product of Powers rule, the x bases and the y bases are handled separately to simplify to \(x^4y^4\).

Multiplying Polynomial Terms

Multiplying polynomial terms might seem daunting, but it's really about combining like terms and applying exponent rules. When you have two polynomials, you distribute each term in the first polynomial across each term in the second polynomial, also known as the FOIL method when dealing with binomials.

Consider the following format for a basic polynomial multiplication: \(a * b\), where \(a\) and \(b\) are the terms with coefficients and variables. Multiply the coefficients (numerical parts) with each other, multiply the variables using the laws of exponents, then combine the results.

Example:

If we have \(12xy^2\) and \( -2x^3y^2 \), we multiply the coefficients: \(12 * -2 = -24\). Next, we multiply the variable parts: \(x * x^3 = x^4\) and \(y^2 * y^2 = y^4\). The final result is putting these together to get: \( -24x^4y^4\).

Algebraic Expressions Simplification

The goal of simplifying algebraic expressions is to make them as straightforward as possible, ideally with fewer terms and lower exponents. This process often involves combining like terms, using the distributive property, factoring, and canceling.

The key steps to simplifying an algebraic expression include:

• Combining like terms, which have the same variable parts
• Applying the distributive property to remove parentheses
• Factoring out common factors between terms
• Canceling out terms where possible, especially in fractions

Following these steps leads to a much clearer expression, which can be crucial for solving equations or understanding the behavior of functions. In our exercise, after applying the laws of exponents and the multiplication of polynomial terms, the expression is already in its simplest form, \( -24x^4y^4 \), with no like terms left to combine or further rules to apply.