Problem 65 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{\left(u^{-1} v^{2}\right)^{2}}{\left(u^{3} v^{-2}\right)^{3}} $$

Step-by-Step Solution

Answer

\(\frac{v^{10}}{u^{11}}\)

Step 1: Expand the Numerator

The numerator of the given fraction is \((u^{-1} v^{2})^2\). Use the Power Rule of Exponents: \((a^m)^n = a^{m \cdot n}\). Apply this to each variable to get: \(u^{-1 \cdot 2} v^{2 \cdot 2} = u^{-2} v^{4}\).

Step 2: Expand the Denominator

The denominator is \((u^3 v^{-2})^3\). Again, use the Power Rule: \((a^m)^n = a^{m \cdot n}\). Apply this to each variable to get: \(u^{3 \cdot 3} v^{-2 \cdot 3} = u^{9} v^{-6}\).

Step 3: Divide the Expressions

Divide the expanded numerator by the expanded denominator: \(\frac{u^{-2} v^{4}}{u^{9} v^{-6}}\). Use the Quotient Rule for exponents: \(\frac{a^m}{a^n} = a^{m-n}\). Apply this to each variable: \(u^{-2-9} v^{4-(-6)} = u^{-11} v^{10}\).

Step 4: Remove Negative Exponent

The expression we have is \(u^{-11} v^{10}\). According to the rule of negative exponents, \(a^{-n} = \frac{1}{a^n}\). Rewrite \(u^{-11}\) as \(\frac{1}{u^{11}}\), giving us: \(\frac{1 \cdot v^{10}}{u^{11}} = \frac{v^{10}}{u^{11}}\).

Key Concepts

Power Rule of Exponents

The power rule of exponents is a fundamental tool in simplifying expressions, especially when dealing with exponents raised to another power. The rule can be stated as:

• \((a^m)^n = a^{m \cdot n}\)

This means that when you have an exponent raised to another exponent, you multiply the exponents together.
For example, in the expression \((u^{-1} v^{2})^2\), we apply the power rule separately to each variable:

• For \(u^{-1}\), it becomes \(u^{-1 \cdot 2} = u^{-2}\).
• For \(v^{2}\), it changes to \(v^{2 \cdot 2} = v^{4}\).

Using this rule simplifies the complexity of expressions and sets the stage for further simplification.

Quotient Rule for Exponents

The quotient rule for exponents helps in simplifying expressions where like bases are being divided. This rule is expressed as:

• \(\frac{a^m}{a^n} = a^{m-n}\)

Simply put, when dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
For example, when we simplify \(\frac{u^{-2} v^{4}}{u^{9} v^{-6}}\), the quotient rule helps us reduce it:

• For \(u\), it becomes \(u^{-2-9} = u^{-11}\).
• For \(v\), it is \(v^{4 - (-6)} = v^{4+6} = v^{10}\).

This rule enables handling complex fractions much more efficiently.

Negative Exponents

Understanding negative exponents is crucial in simplifying algebraic expressions. A negative exponent indicates a reciprocal. For any nonzero \(a\), the rule is:

• \(a^{-n} = \frac{1}{a^n}\)

This means, if you encounter a negative exponent, flip the base to the denominator and change the exponent to positive.
Applying this to our previous result, \(u^{-11}\) can be rewritten as \(\frac{1}{u^{11}}\). This transformation is essential for expressing results with only positive exponents. The final simplified expression is then transformed to \(\frac{v^{10}}{u^{11}}\), making it clearer and easier to interpret.

Algebraic Fractions

Algebraic fractions involve expressions with variables in the numerator, denominator, or both. Simplifying them often includes

• Expanding powers using the power rule.
• Applying the quotient rule for exponents.
• Eliminating negative exponents by rewriting them with positive exponents.

In our example, the expression \(\frac{(u^{-1} v^{2})^2}{(u^{3} v^{-2})^3}\) was simplified by applying these procedures.
Ensuring that the final form has only positive exponents is vital because it often represents more natural or standard forms in algebraic work. This technique is universal whether dealing with variables or real numbers, enabling you to simplify complex algebraic expressions effectively.