Problem 50 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\frac{5 x y^{-2}}{x^{-1} y^{-3}}\) (b) \(\left(\frac{2 a^{-1} b}{a^{2} b^{-3}}\right)^{-3}\)

Step-by-Step Solution

Answer

(a) 5x²y, (b) \(\frac{a^9}{8b^{12}}\).

Step 1: Simplify Expression (a)

Start with the expression \( \frac{5 x y^{-2}}{x^{-1} y^{-3}} \). Apply the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \) for both \( x \) and \( y \):\[ \frac{5 x y^{-2}}{x^{-1} y^{-3}} = 5 x^{1 - (-1)} y^{-2 -(-3)} = 5 x^{2} y^{1}.\] After simplification, the expression is \( 5 x^2 y \).

Step 2: Simplify Expression with Negative Exponents (b)

Consider the expression \( \left(\frac{2 a^{-1} b}{a^{2} b^{-3}}\right)^{-3} \). First, simplify the expression inside the parenthesis by applying the exponent rule \( \frac{a^m}{a^n} = a^{m-n} \):\[ \frac{2 a^{-1} b}{a^{2} b^{-3}} = 2 a^{-1-2} b^{1-(-3)} = 2 a^{-3} b^{4}. \] Now raise the entire fraction to the power of \(-3\):\[ \left(2 a^{-3} b^{4}\right)^{-3} = 2^{-3} a^{9} b^{-12}.\] Simplify each part: \( 2^{-3} = \frac{1}{8} \), so the expression is \( \frac{1}{8} a^{9} b^{-12} \).

Step 3: Eliminate Negative Exponents (b)

Convert negative exponents to positive exponents by moving them between the numerator and the denominator. From \( \frac{1}{8} a^{9} b^{-12} \), moving \( b^{-12} \) gives: \[ \frac{a^{9}}{8 b^{12}}. \] Now the expression has no negative exponents.

Key Concepts

Simplifying Expressions

Simplifying an expression involves making it as simple as possible by combining like terms and reducing any fractions. This process often includes eliminating negative exponents to ensure a more "standard" form of expression. Simplification makes expressions easier to read and understand. In the given examples, the expressions are simplified by using the rules of exponents.

• First, identify the parts of the expression that can be combined or reduced.
• Apply relevant mathematical laws like exponent rules or fraction simplification.
• Rewrite the expression in its simplest form, removing any negative exponents or simplifying fractions whenever possible.

As demonstrated in original Exercise (a), the expression was simplified to get rid of negative exponents, resulting in a simple term that's easy to interpret.

Exponent Rules

Exponent rules are essential in simplifying expressions, especially when dealing with powers and roots. The main rules involve adding, subtracting, and multiplying exponents correctly. For example:

• The product of powers rule: \[ a^m \times a^n = a^{m+n} \]
• The power of a power rule: \[ (a^m)^n = a^{m\times n} \]
• The quotient of powers rule: \[ \frac{a^m}{a^n} = a^{m-n} \]

In the exercises, these rules were applied to merge like bases and simplify the expressions. It's important to follow each rule precisely to avoid mistakes, such as confusing the addition and subtraction of exponents or misapplying the power rule. Correct usage ensures an accurate and simplified result of complex algebraic expressions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Simplifying algebraic fractions involves reducing them to their lowest terms similar to numerical fractions. Key points include:

• Identify any common factors in the numerator and denominator.
• Utilize the rules of exponents to simplify fractions with variables.
• Convert all negative exponents to positive by flipping their position in the fraction.

For example, Exercise (a) was simplified by canceling terms and flipping variables with negative exponents. This approach helps make the algebraic fractions manageable and easier to solve, especially in more complex scenarios involving many variables.

Mathematical Notation

Mathematical notation is a symbolic language of formulas and symbols used to convey mathematical ideas clearly. Proper notation is crucial for understanding and communicating mathematical solutions. When dealing with expressions and exponents, consistent use of notation helps avoid confusion.

• Ensure the correct use of brackets and parentheses to denote operations appropriately.
• Write exponents clearly to denote power, especially when they involve negative numbers.
• Consistently write fractions with a clear numerator and denominator division.

In the step by step solutions provided, correct notation makes it easier to follow and understand the logical flow of simplifying the expressions. As seen in Exercise (b), proper notation was key to track changes in exponents through each step, ultimately leading to an expression free of negative exponents.