Problem 75 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{a^{-5} a^{-9}}{a^{-8}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{a^6}\).

Step 1: Apply the Rule for Multiplying Exponents

When multiplying powers with the same base, add the exponents. Thus:\[a^{-5} \cdot a^{-9} = a^{-5 + (-9)} = a^{-14}.\]

Step 2: Set up the New Fraction

Replace the numerator of the original fraction with the result from Step 1:\[\frac{a^{-14}}{a^{-8}}.\]

Step 3: Apply the Quotient Rule for Exponents

When dividing powers with the same base, subtract the exponents:\[\frac{a^{-14}}{a^{-8}} = a^{-14 - (-8)} = a^{-14 + 8} = a^{-6}.\]

Step 4: Rewrite with Positive Exponents

Express the answer without negative exponents:\[a^{-6} = \frac{1}{a^6}.\]

Key Concepts

Exponents

Exponents are a fundamental mathematical concept. They express how many times a number, known as the base, is multiplied by itself. If you see a number written as \(a^n\), the base is \(a\) and the exponent is \(n\). This means you multiply \(a\) by itself \(n\) times. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). Exponents allow us to write large numbers succinctly and perform calculations more easily.

• Base: The number that is being multiplied.
• Exponent: The number which denotes how many times the base is used as a factor.
• Example: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Understanding exponents is crucial because they appear throughout different areas of mathematics, including algebra, calculus, and beyond.

Negative Exponents

Negative exponents might seem daunting at first, but they are simple once you understand their meaning. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. In other words, \(a^{-n}\) means \(\frac{1}{a^n}\). Negative exponents invert the base and change the exponent to positive. This concept helps simplify expressions and solve equations.
Here’s a breakdown:

• \(a^{-1} = \frac{1}{a}\)
• \(a^{-3} = \frac{1}{a^3}\)
• \(a^{-5} = \frac{1}{a^5}\)

Mastering negative exponents will broaden your understanding of more complex algebraic expressions.

Exponent Rules

Exponent rules make it easier to work with expressions involving powers. These rules include ways to multiply, divide, and power exponents.

• Product Rule: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent Rule: Any non-zero base raised to the power of zero equals one: \(a^0 = 1\), where \(a eq 0\).

By applying these exponent rules, you can simplify expressions effectively and avoid pitfalls when dealing with complex calculations.