Problem 51 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(a^{3} b^{-3} c^{-2}\right)^{-5}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{b^{15} c^{10}}{a^{15}}\).

Step 1: Apply the Power Rule

When raising an exponential expression to a power, multiply the exponents. Apply this rule to each term inside the parentheses:\[(a^3 b^{-3} c^{-2})^{-5} = a^{3 imes (-5)} b^{-3 imes (-5)} c^{-2 imes (-5)}\]This simplifies to:\[a^{-15} b^{15} c^{10}\]

Step 2: Simplify Negative Exponents

Rewrite the expression so that none of the terms have negative exponents. Negative exponents mean reciprocal, so a term with a negative exponent should be moved to the denominator:\[a^{-15} b^{15} c^{10} = \frac{b^{15} c^{10}}{a^{15}}\]

Step 3: Final Simplified Form

Ensure the expression has no negative exponents, and all terms are in their simplest form. The expression \(\frac{b^{15} c^{10}}{a^{15}}\) already meets these conditions. Thus, this is the final simplified version of the expression.

Key Concepts

Power Rule

The power rule is a fundamental concept in algebra that helps simplify expressions involving exponents. Whenever you have an expression raised to an additional power, the power rule says to multiply the exponents. This rule is very efficient as it converts complex expressions into manageable ones.
Let's take the expression \((a^m b^n c^p)^q\). According to the power rule, you multiply the exponents inside the expression by the outer exponent, resulting in \(a^{m \times q} b^{n \times q} c^{p \times q}\). This method is applied to consolidate an expression raised to another power, making it more straightforward for further simplification.
Consider simplifying the expression \((a^3 b^{-3} c^{-2})^{-5}\). Apply the power rule to get each component: \(a^{3 \times (-5)}\), \(b^{-3 \times (-5)}\), and \(c^{-2 \times (-5)}\). After multiplication, the expression becomes \(a^{-15} b^{15} c^{10}\). Using the power rule effectively is crucial to prevent mistakes and ensure accurate simplification.

Negative Exponents

Exponents can be positive or negative, and understanding the rules for negative exponents is important for simplifying expressions. A negative exponent indicates that the base needs to be inverted as a fraction. In simple terms, it tells us to move the base to the opposite side of the fraction line.
For instance, if you have \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). Conversely, \(\frac{1}{a^{-n}}\) becomes \(a^n\). This reciprocal relationship is how we turn negative exponents into positive ones.
In the expression \(a^{-15} b^{15} c^{10}\), applying the negative exponent rule, \(a^{-15}\) converts to \(\frac{1}{a^{15}}\), thus shifting the base \(a\) from the numerator to the denominator. The expression becomes \(\frac{b^{15} c^{10}}{a^{15}}\), which removes all negative exponents and harmonizes the expression, ensuring consistency.

Simplification of Expressions

Simplifying expressions is a crucial skill in algebra that involves reducing expressions to their most concise form. It ensures that calculations are easier to handle and interpret, and it removes unnecessary complexity.
Begin by applying the power rule and addressing any negative exponents as shown in earlier steps. Using these rules, we turn a complicated expression like \((a^3 b^{-3} c^{-2})^{-5}\) into a clear fraction \(\frac{b^{15} c^{10}}{a^{15}}\).
This expression no longer has negative exponents and is in its simplest form, making it easy to work with in further mathematical operations. Simplification allows expressions to be accurately compared, substituted, or calculated, laying the foundation for more complex algebraic manipulations.