Problem 23 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(a^{3} b^{3}\right)(a b)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(a b\).

Step 1: Apply the Power of a Power Rule

Recall that the power of a power rule tells us that \(x^m imes x^n = x^{m+n}\). Applying this rule to the expression \((a^3 b^3)(a b)^{-2}\):\[a^3 b^3\times a^{-2} b^{-2}\] gives us \[a^{3+(-2)} b^{3+(-2)}\]. This simplifies to \[a^{1} b^{1}\].

Step 2: Simplify Further

Since the exponents 1 on both \(a\) and \(b\) imply that no exponentiation is actually necessary, we can simplify the expression \[a^{1} b^{1}\] to just \[a b\].

Key Concepts

Exponent Rules

Exponent rules are fundamental to simplifying expressions involving powers. They help us easily manipulate terms with exponents without expanding everything fully. Here are some of the key rules:

• Product of Powers Rule: When multiplying two exponents with the same base, you add the exponents. For example, \(x^m \times x^n = x^{m+n}\).
• Quotient of Powers Rule: When dividing, subtract the exponents of the same base: \(\frac{x^m}{x^n} = x^{m-n}\).
• Zero Exponent Rule: Any non-zero number raised to the zero power is 1, such as \(x^0 = 1\).
• Negative Exponent Rule: A negative exponent indicates reciprocal. For instance, \(x^{-m} = \frac{1}{x^m}\).

Learning these rules helps transform complex expressions into simpler ones, making calculations much more approachable.
In this exercise, knowing the product and negative exponent rules simplifies the problem quickly.

Power of a Power Rule

The "power of a power rule" is a specific exponent rule focused on expressions where an exponent is taken to another power. It states that when you have a power raised to another power, you multiply the exponents. Mathematically, this is represented as \((x^m)^n = x^{m \cdot n}\).
This rule allows simplification of expressions that look more complicated at first glance. Instead of working through each level of multiplication, you can simplify expressions more directly by manipulating the exponents.
In the given problem, while the main focus is applying the product of powers, it's essential to recognize that this same concept of multiplying exponents applies when you encounter nested powers. Understanding it can help prevent mistakes and miscalculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). These expressions can appear complex, but they become manageable with the right techniques and strategies.
To work with algebraic expressions efficiently:

• Identify like terms: These are terms that contain the same variables raised to the same power. Combine them by adding or subtracting their coefficients.
• Simplify using algebraic rules: Apply exponent rules, distribution, and factoring as needed to make expressions simpler.
• Check for special cases: Recognize patterns like difference of squares \(a^2 - b^2 = (a-b)(a+b)\) or perfect square trinomials \((a \pm b)^2\).

The primary goal is to rewrite these expressions in a form that's easier to work with or solve, as seen in the provided exercise, where initially complex exponents were simplified to express the solution as \(a b\).