Problem 58 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(\frac{6 a b^{4}}{3 x^{-3} y^{-4}}\right)^{-1} $$

Step-by-Step Solution

Answer

\( \frac{1}{2 a b^{4} x^{3} y^{4}} \)

Step 1: Rewrite the Expression with Positive Exponents

Start with the given expression: \( \left(\frac{6 a b^{4}}{3 x^{-3} y^{-4}}\right)^{-1} \). Inside the fraction, move the terms with negative exponents in the numerator and denominator to the opposite part of the fraction to make their exponents positive. This gives: \( \frac{6 a b^{4} x^{3} y^{4}}{3} \).

Step 2: Simplify the Fraction

Now, simplify the fraction \( \frac{6 a b^{4} x^{3} y^{4}}{3} \). Divide both the numerator and the denominator by 3: \( 6 \div 3 = 2 \). This simplifies to \( 2 a b^{4} x^{3} y^{4} \).

Step 3: Apply the Negative Exponent

The original expression has an overall negative exponent: \( (2 a b^{4} x^{3} y^{4})^{-1} \). Applying the rule of negative exponents \( a^{-n} = \frac{1}{a^{n}} \), rewrite the expression as: \( \frac{1}{2 a b^{4} x^{3} y^{4}} \).

Key Concepts

Understanding Exponents

Exponents are a fundamental part of algebra that describe how many times a number, known as the base, is multiplied by itself. For example, in the expression \( a^n \), \( a \) is the base, and \( n \) is the exponent, indicating that \( a \) is multiplied by itself \( n \) times. Here are a few key points about exponents:

• Positive Exponents: These represent repeated multiplication. For instance, \( 3^2 = 3 \times 3 \).
• Zero Exponents: Any base raised to the power of zero is 1, such as \( a^0 = 1 \), given that \( a eq 0 \).
• Negative Exponents: These signify division, represented as the reciprocal of the base raised to the positive of the exponent, like \( a^{-n} = \frac{1}{a^n} \).

Grasping the concept of exponents is crucial for simplifying expressions, as you'll often need to apply these rules to manipulate and reduce expressions to their simplest forms.

Simplifying Algebraic Expressions

Simplifying expressions involves making them as compact as possible while retaining the same value. In algebra, this often means eliminating parentheses, combining like terms, reducing fractions, and changing negative exponents to positive ones. The main goal is to make an expression easy to understand and work with. Here are steps to simplify algebraic expressions:

• Combine Like Terms: Terms that have the same variable and exponent. For example, \( 3x^2 + 2x^2 = 5x^2 \).
• Apply Distributive Property: When you have an expression like \( a(b + c) \), distribute to get \( ab + ac \).
• Convert Negative Exponents: Make all exponents positive by rewriting terms with negative exponents in the denominator. For example, \( x^{-2} \) becomes \( \frac{1}{x^2} \).

Understanding these steps can help tackle expressions, turning complex algebraic statements into simpler, more manageable forms.

Working with Negative Exponents

Negative exponents might look intimidating at first, but they're just a different way to represent fractions. The negative exponent rule states \( a^{-n} = \frac{1}{a^n} \), meaning any term with a negative exponent can be rewritten as a fraction with a positive exponent. When simplifying expressions with negative exponents:

• Flip and Make Positive: Move terms with negative exponents between numerator and denominator to make them positive. For example, in \( x^{-3} \), it becomes \( \frac{1}{x^3} \) if moved to the denominator.
• Consistency: Remember that this rule applies at all levels of algebra, whether dealing with simple numbers or complex expressions.

By understanding and applying this rule, you'll find it much easier to work through challenging algebraic problems. Keeping track of where terms are (numerator vs. denominator) ensures you don't lose track when rewriting and solving these expressions.