Problem 62 - College Algebra

Question

Simplify each exponential expression $$ \left(\frac{-30 a^{14} b^{8}}{10 a^{17} b^{-2}}\right)^{3} $$

Step-by-Step Solution

Answer

\(-27 a^{-9} b^{30}\)

Step 1: Simplify the fraction and exponents

The fraction and the exponent in the given term have to be simplified first. \(-30/10\) simplifies to \(-3\) and \(a^{14}/a^{17}\) simplifies to \(a^{-3}\) (using the exponent rule \(a^{m}/a^{n} = a^{m-n}\). Similarly, \(b^{8}/b^{-2}\) simplifies to \(b^{10}\) (using the same exponent rule). The given term then becomes \[(-3a^{-3}b^{10})^{3}\]

Step 2: Apply exponent rule to the simplified term

Now, apply the exponent rule \((ab)^n = a^n * b^n\) to the simplified term. Therefore the term becomes \[(-3)^3 * (a^{-3})^3 * (b^{10})^3\]

Step 3: Simplify the resulting expression

The final step is to perform the powering operation and simplify the variables. \((-3)^3\) becomes \(-27\), \((a^{-3})^3\) simplifies to \(a^{-9}\) and \((b^{10})^3\) simplifies to \(b^{30}\). Thus, the resulting term is \[-27 a^{-9} b^{30}\]

Key Concepts

Simplifying Fractions in Exponential Expressions

When dealing with expressions that include fractions, it's important to simplify them as much as possible before applying other mathematical operations. In an expression like \(\left(\frac{-30 a^{14} b^{8}}{10 a^{17} b^{-2}}\right)^{3}\), the first step would be to simplify the fraction itself. This means reducing the coefficients as well as any algebraic terms found in the numerator and denominator.

• Reduce Coefficients: Begin by simplifying the numerical coefficients. Dividing \(-30\) by \(10\) gives \(-3\).
• Apply Exponent Rules: Use the rule \(a^{m}/a^{n} = a^{m-n}\). For this exercise, \(a^{14}/a^{17} = a^{-3}\) and \(b^{8}/b^{-2} = b^{10}\).

After simplifying, the fraction is transformed to \(-3a^{-3}b^{10}\). This step sets the stage for applying further exponent rules.

Understanding and Applying Exponent Rules

Exponent rules are foundational when simplifying expressions that include powers. Let's explore some key rules and see how they are applied in the given expression.

• Division of Powers: When dividing like bases, subtract the exponents. This is shown in \(a^{14}/a^{17} = a^{-3}\). Similarly for \(b^{8}/b^{-2} = b^{10}\).
• Power of a Power: When raising a power to another power, multiply the exponents. For example, \((a^{-3})^3 = a^{-9}\) because \(-3 \times 3 = -9\).
• Power of a Product: Distribute the power to every term in the product. In \((-3a^{-3}b^{10})^3\), compute each part separately as \((-3)^3\), \((a^{-3})^3\), and \((b^{10})^3\).

Applying these rules transforms the expression step by step, simplifying it and preparing it for final calculation.

Breaking Down Algebraic Expressions

Algebraic expressions, especially those involving multiple variables and powers, can look intimidating at first. Breaking them down makes them manageable.Here, the expression \((-3a^{-3}b^{10})^3\) initially incorporates coefficients, variables and exponents:

• Coefficients: This is the numeric part, \(-3\). Raising it to a power, \((-3)^3\), results in \(-27\).
• Variables with Exponents: These are variables like \(a\) and \(b\) raised to certain powers. Using power and product rules, \((a^{-3})^3 = a^{-9}\) and \((b^{10})^3 = b^{30}\).

When broken down, algebraic expressions become simple, leading to easier computation and a clearer understanding of each part's role in the final result.