Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[6]{\left(2 u^{-3} v^{4}\right)^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2v^4}{u^3} \).

Step 1: Apply the Power of a Power Rule

The expression given is \( \sqrt[6]{(2u^{-3}v^4)^6} \). According to the power of a power rule, \((a^m)^n = a^{m \cdot n}\), so we apply this inside the sixth root: \( (2u^{-3}v^4)^6 = 2^6 u^{-18} v^{24} \).

Step 2: Simplify the Sixth Root

Now the expression inside the root is \( \sqrt[6]{2^6 u^{-18} v^{24}} \). The sixth root and the sixth power cancel out, as \( \sqrt[6]{a^6} = a \). So, we have: \( 2 u^{-3} v^4 \).

Step 3: Rationalizing the Expression

Currently, the expression is \( 2 u^{-3} v^4 \). Since \( u^{-3} \) is in the denominator, we convert it to the numerator by writing it as \( \frac{2v^4}{u^3} \).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is a handy tool in simplifying expressions where exponents are involved inside a parenthesis. Imagine you have something like \((a^m)^n\). The rule says you can simplify this by multiplying the exponents together, so it becomes \(a^{m \cdot n}\). For example, if you encounter \((x^2)^3\), applying the rule gives you \(x^{2 \cdot 3} = x^6\).

This concept makes handling large expressions more manageable. It reduces the complexity by eliminating extra layers. In the original problem, we used this rule to transform \((2u^{-3}v^4)^6\) into \(2^6 u^{-18} v^{24}\). Each part of the expression is raised to the power of 6 individually, giving us a simplified base to work with.

• Remember: Multiply the exponents when raising a power to another power.
• This helps to streamline expressions and clear up nested exponents.

Rationalizing the Denominator

Rationalizing the denominator is a method used in algebra that involves rewriting an expression so that no radicals (such as square roots) appear in the denominator. For our problem, after simplifying the root, the expression transformed into \(2u^{-3}v^4\). In this form, \(u^{-3}\) implies \(u^3\) is in the denominator, which is not the conventional way to present an answer.

To rationalize, we rewrite the expression as \(\frac{2v^4}{u^3}\). This moves the variable with a negative exponent from the numerator to the denominator, where it naturally resides in positive form. This step improves the clarity of the expression and aligns with standard mathematical practices.

• Rationalizing the denominator clears up expressions by removing radicals where they aren't easily managed.
• This technique also involves moving expressions with negative exponents to their appropriate position in the fraction.

Roots and Radicals

Roots and radicals are foundational to many areas of mathematics. They help us solve equations involving powers and exponents. The sixth root like \(\sqrt[6]{a}\), indicates a number that, when raised to the power of six, equals \(a\). This is similar to more common roots like square roots or cube roots but applies to the sixth power.

In the given exercise, at one point, the expression reduced to the sixth root of a perfect sixth power, \(\sqrt[6]{2^6 u^{-18} v^{24}}\). Because \(\sqrt[6]{a^6} = a\), the root and the power cancel each other out, leaving us with \(2 u^{-3} v^4\).

• The concept of roots allows us to reverse-engineer powers, essentially "undoing" them to reach a base value.
• Understanding how roots interact with powers, like when they cancel each other out, simplifies complicated expressions thoroughly.