Problem 23 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(-3 a^{-5}\right)\left(5 a^{-7}\right)$$

Step-by-Step Solution

Answer

\(-15a^{-12}\)

Step 1: Rewrite the expression with positive exponents

First, let's rewrite the given expression with positive exponents: $$\left(-3 a^{-5}\right)\left(5 a^{-7}\right) = \frac{-3}{a^5} \cdot \frac{5}{a^7}$$

Step 2: Multiply the constants together

Since there are no parentheses indicating otherwise, we can multiply the constants directly: $$(-3) \cdot (5) = -15$$

Step 3: Multiply the expressions with the same base

We can now multiply the expressions with the same base "a", taking into account the base's respective exponents. According to exponent multiplication rule, when multiplying two expressions with the same base, we add the exponents: $$\frac{1}{a^5} \cdot \frac{1}{a^7} = \frac{1}{a^{5+7}} = \frac{1}{a^{12}}$$

Step 4: Rewrite the expression in the simplest form

Combine the result from Step 2 and Step 3 to rewrite the expression in the simplest form: $$-15 \cdot \frac{1}{a^{12}} = \boxed{-15a^{-12}}$$

Key Concepts

Understanding Negative Exponents

In mathematics, exponents offer a convenient way to denote repeated multiplication. Negative exponents can appear confusing, but they follow a simple rule: a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, if we have \(a^{-5}\), it means \(\frac{1}{a^5}\). The negative sign doesn't mean negative in value, but rather shifts the base from the numerator to the denominator. Here’s what you should remember:

• \(a^{-n} = \frac{1}{a^n}\)
• The reciprocal rule converts negative exponents to positive.
• This rule helps in simplifying expressions by turning them into products and fractions for easier handling.

Recognizing negative exponents and converting them into positive terms is often the first step in simplifying expressions, which is critical for further operations like multiplication or division of powers.

Applying Exponent Rules

Exponent rules, also known as the laws of exponents, are foundational in algebra for manipulating expressions efficiently. These rules include multiplying powers, dividing powers, raising a power to another power, and handling exponents of zero. In our example, we focus on the product of powers rule, which states: when multiplying two exponents with the same base, you can add the exponents. Thus, for \(a^m \times a^n = a^{m+n}\). Some key exponent rules include:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)
• Power of a product: \((ab)^n = a^n \times b^n\)

Using these rules simplifies complex expressions and prepares them for further calculations.

Steps to Simplify Expressions

Simplifying expressions is a fundamental algebraic skill that makes equations easier to analyze and solve. The goal is to express the given mathematical statement with fewer terms or simpler terms. We use exponent rules extensively in this process, especially when dealing with expressions involving powers. Let’s break down the process of simplification:

• Rewrite negative exponents as positive to use product and quotient rules effectively. This simplifies the expression and clarifies relationships between terms.
• Apply the product of powers rule to combine terms with the same base by adding exponents together.
• Simplify constants separately from the variable terms to ensure clean results.

In the initial problem \((-3a^{-5})(5a^{-7})\), key simplification steps involved converting negative exponents to reciprocals and applying the product of powers rule to sum the exponents. The result, after simplification, became a more manageable expression with fewer components.