Problem 38 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \frac{a^{-3} b^{4}}{a^{-5} b^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a^2}{b} \).

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). So, move terms with negative exponents to the opposite side of the fraction line to make them positive:\[\frac{a^{-3} b^{4}}{a^{-5} b^{5}} = \frac{b^4 \cdot a^5}{b^5 \cdot a^3}\]

Step 2: Simplify the Expression

Now, simplify the expression by applying the laws of exponents. We use the quotient of powers rule, \( \frac{x^m}{x^n} = x^{m-n} \):For \(a\) terms:\[\frac{a^5}{a^3} = a^{5-3} = a^2\]For \(b\) terms:\[\frac{b^4}{b^5} = b^{4-5} = b^{-1} = \frac{1}{b}\]The expression becomes \( a^2 \cdot \frac{1}{b} = \frac{a^2}{b} \).

Step 3: Final Simplified Expression

The final simplified expression after eliminating negative exponents and simplifying is:\[\frac{a^2}{b}\]

Key Concepts

Negative Exponents

Negative exponents might seem a bit intimidating at first. However, they are only a different way to express fractions or reciprocals. The law states that any number with a negative exponent can be rewritten as a fraction, specifically as its reciprocal. For example, the expression \( a^{-n} \) means the same thing as \( \frac{1}{a^n} \). This transformation helps us make negative exponents positive, simplifying calculations. Let's consider \( a^{-3} \). According to the negative exponent rule, \( a^{-3} = \frac{1}{a^3} \). So, when you see a negative exponent, just remember: turn it into a fraction by placing everything under 1, and the exponent will become positive!

Laws of Exponents

The laws of exponents are a powerful set of rules that simplify complex expressions involving powers. The key laws include:

• Product of Powers: \( x^m \times x^n = x^{m+n} \)
• Quotient of Powers: \( \frac{x^m}{x^n} = x^{m-n} \)
• Power of a Power: \((x^m)^n = x^{m\times n}\)

Each rule helps you handle specific situations when dealing with powers of numbers or variables. By applying these laws, you can simplify expressions and make calculations easier. For example, if you have \( x^5 \cdot x^2 \), use the product of powers rule to combine them: \( x^{5+2} = x^7 \). Knowing these rules makes working with algebraic expressions much more manageable.

Quotient of Powers Rule

The quotient of powers rule helps when dividing two numbers or variables with the same base. This rule tells us to subtract the exponent of the denominator from the exponent of the numerator. In essence, if you have \( \frac{x^m}{x^n} \), it simplifies to \( x^{m-n} \). This means you aren't multiplying the numbers but rather, using subtraction.
In our exercise, we applied this rule to both \( a \) and \( b \). For example, with \( \frac{a^5}{a^3} \), the result is \( a^{5-3} = a^2 \). Similarly, \( \frac{b^4}{b^5} \) results in \( b^{4-5} = b^{-1} = \frac{1}{b} \). This rule, when combined with understanding negative exponents, transforms a seemingly complex fraction into a neat and simple expression. It's a handy tool for anyone working with exponents and trying to simplify algebraic fractions.