Problem 44 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(-3 x^{4} y^{6}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(-27 x^{12} y^{18}\).

Step 1: Identify the Expression

We need to evaluate or simplify: \(\left(-3 x^{4} y^{6}\right)^{3}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The simplified form of the given exponential expression is \(-27 x^{12} y^{18}\).

Key Concepts

Exponent Distribution

Simplifying an expression with exponents often involves exponent distribution, which is a technique based on the Power of a Power rule. When you have a power raised to another power, as in \( (a^m)^n \), you multiply the exponents. In our exercise, the term \( (-3 x^4 y^6)^3 \) requires distribution of the exponent 3 to each component inside the parentheses.

This means that we need to apply the power to the numerical coefficient and each variable separately. We calculate the cube of -3 and similarly, multiply the exponents for both \( x \) and \( y \) by 3. The process is straightforward but requires attentiveness to operations involving negative numbers and exponent multiplication. In effect, distributing the exponent streamlines the expression into a simpler form by breaking it down sequentially and applying basic arithmetic operations.

Using this approach ensures that students can handle more complex expressions involving variables and coefficients raised to higher powers.

Exponential Expression Simplification

After exponent distribution, the focus shifts to exponential expression simplification. This involves combining like terms and reducing expressions to their simplest form. In the context of the given exercise, once we've applied the exponent to each term, we end up with \( -27 x^{12} y^{18} \) as the distributed expression.

To simplify further, if there were similar terms or factors that could be combined or reduced, that would be the next step. Here, however, we already have the expression in its simplest form because there are no like terms to be combined.

One key aspect to remember is to watch for special cases, such as zero exponents or negative exponents, which can significantly alter the form of the simplified expression. Ensuring that students remember these special cases is crucial for mastering simplification of exponential expressions.

Exponential Rules

Grasping exponential rules is pivotal in simplifying exponential expressions. These rules provide us with a set of tools to handle various exponent-related operations. The exercise we're examining employs one of the main rules, the Power of a Product/Power of a Power rule, which states that when taking the power of a product, you distribute the exponent to all factors inside the product.

Other important rules include:

• Product of Powers: When multiplying like bases, add the exponents, as in \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, subtract the exponents, as in \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent: Any base raised to the power zero equals one, as in \( a^0 = 1 \) (assuming \( a eq 0 \)).
• Negative Exponent: Indicates the reciprocal of the base raised to the positive exponent, as in \( a^{-n} = \frac{1}{a^n} \) (again, assuming \( a eq 0 \)).

Understanding these rules and knowing when and how to apply them is fundamental for algebraic proficiency.

Algebraic Manipulation

The final concept within our problem-solving toolbox is algebraic manipulation. In dealing with exponential expressions, it's crucial to be skilled in manipulative techniques that involve rearrangement and reformation of algebraic equations or expressions.

Manipulation encompasses various operations, which may include distributing exponents, factoring, expanding, and simplifying expressions. It's an essential skill set that allows for a broader understanding and solving of algebraic problems. The better one is at manipulating algebraic expressions, the easier it becomes to break down complex problems into simpler, solvable components.

As seen in the given exercise, the manipulation involves distributing an exponent across terms within the parentheses, transforming an initially daunting expression into a more manageable one. This pivotal ability lays the groundwork for success in more advanced areas of mathematics, as it forms the cornerstone of algebraic problem-solving.