Problem 63 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{8 a^{3} b^{-4}}{2 a^{-5} b^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4a^8}{b^9} \).

Step 1: Simplify the Fraction

Start by simplifying the coefficient part of the fraction: \( \frac{8}{2} = 4 \). Now you have \( 4 \frac{a^{3} b^{-4}}{a^{-5} b^{5}} \).

Step 2: Apply the Law of Exponents

For the \(a\) terms in the fraction, use the property \( a^m / a^n = a^{m-n} \):\[ a^{3 - (-5)} = a^{3 + 5} = a^{8} \]. For the \(b\) terms, we use \( b^{-4} / b^{5} = b^{-4-5} = b^{-9} \).

Step 3: Eliminate Negative Exponents

Rewrite \( b^{-9} \) using positive exponents: \( b^{-9} = \frac{1}{b^{9}} \). Therefore, the expression is now: \[ 4 a^{8} \frac{1}{b^{9}} = \frac{4 a^{8}}{b^{9}} \].

Key Concepts

Exponents Rules

Understanding the rules of exponents is essential in simplifying algebraic expressions. Exponents, also known as powers, represent repeated multiplication of a base number. When dealing with exponents, it is crucial to grasp the basic laws that make manipulation and simplification easier. Here are some key exponent rules:

• Product of Powers Rule: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \( a^m / a^n = a^{m-n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

By applying these rules, we simplify complex expressions efficiently. In this exercise, we used the quotient of powers rule to simplify terms like \( a^3 / a^{-5} \) and \( b^{-4} / b^{5} \), making the expression more manageable.

Negative Exponents

Negative exponents can sometimes seem intimidating, but they are just another way to express division or reciprocals. A negative exponent indicates that the base is on the wrong side of a fraction, and you need to flip it to the opposite side.

• The expression \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \).
• Similarly, \( \frac{1}{b^{-n}} \) is equal to \( b^n \).

In the given problem, we encounter \( b^{-4} \) and \( b^{-9} \). To "eliminate" the negative exponents, we rewrite \( b^{-9} \) as \( \frac{1}{b^9} \). This technique helps transform complex expressions into simpler, more recognizable forms. Understanding how to manage negative exponents is a valuable skill in algebra, ensuring a clear path to the final solution.

Fraction Simplification

Simplifying fractions is a core aspect of algebra that involves reducing expressions to their most basic form. It often requires using exponent rules to combine or reduce terms. Here are some steps to simplify fractions effectively:

• Simplify the numerical coefficients by dividing them: \( \frac{8}{2} = 4 \).
• Apply the exponent rules to simplify variable terms.

In our example, we started by simplifying the coefficient to get a cleaner expression. Then, by applying the quotient of powers rule, the expression \( \frac{a^3}{a^{-5}} \) became \( a^8 \). After handling the negative exponents, the simplified expression is \( \frac{4a^8}{b^9} \). This process of fraction simplification ensures that the final result is easier to interpret and work with, helping to avoid any potential errors in calculations or further problem-solving steps.