Problem 76 - Precalculus: Functions and Graphs

Question

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 expression simplifies and rationalizes to \(\frac{2v^4}{u^3}\).

Step 1: Evaluate the Inner Exponentiation

Start by simplifying the expression inside the radical: \((2 u^{-3} v^{4})^{6}\). To do this, apply the power to each term inside the parentheses: \[2^{6} (u^{-3})^{6} (v^{4})^{6} = 2^{6} u^{-18} v^{24}\]

Step 2: Simplify the Sixth Root

Now, take the sixth root of the entire expression \(2^6 u^{-18} v^{24}\). Since it is under a sixth root, simplify it as follows:\[\sqrt[6]{2^6 u^{-18} v^{24}} = 2^{6/6} u^{-18/6} v^{24/6}\]

Step 3: Calculate Each Term

Calculate the simplified expression term by term:\[2^{6/6} = 2^1 = 2\]\[u^{-18/6} = u^{-3}\]\[v^{24/6} = v^4\]

Step 4: Combine the Results

Combine the results from the previous step to fully simplify the expression:\[2 u^{-3} v^4\]

Step 5: Rationalize the Expression

Since the expression has a negative exponent in \(u^{-3}\), rationalize it by rewriting it with positive exponents:\[2 \cdot \frac{v^4}{u^3}\]

Step 6: Final Simplified Expression

The final simplified and rationalized expression is:\[\frac{2v^4}{u^3}\]

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. - When you see something like \(x^n\) in math, it means multiplying the base \(x\) by itself \(n\) times. - If the exponent is a positive integer, it tells you how many times to multiply the number by itself. - For negative exponents, it indicates division or reciprocal. For example, \(x^{-n} = \frac{1}{x^n}\). In the exercise, we start with \( (2 u^{-3} v^4)^{6} \). Here, each part of the expression inside the parentheses is raised to the 6th power. This means multiplying the exponents:- For the numerical base 2: \(2^6\) simply means multiplying 2 by itself six times.- For the variable \(u^{-3}\), raising to the 6th power becomes \(u^{-18}\) because you multiply \(-3 imes 6\). - Similarly, \(v^4\) becomes \(v^{24}\).These principles help transform the expression by breaking down more complex operations into simpler multiplications.

Radicals and Roots

Radicals and roots simplify complex expressions, allowing us to work with and understand large or complex numbers more effectively. - The symbol \( \sqrt[n]{x} \) denotes the nth root of x, which means we are finding a number that, when raised to the \(n\)-th power, returns \(x\).In our exercise, we have to find the sixth root of the expression \(2^6 u^{-18} v^{24}\), - Finding the sixth root involves dividing the exponents by 6. \( - 2^{6/6} = 2^1 = 2\) - \(u^{-18/6} = u^{-3}\) - \(v^{24/6} = v^4\)- Each term's exponent simplifies by dividing by 6, clarifying the expression neatly.Understanding how roots work is crucial for managing and simplifying algebraic expressions, particularly involving fractional or complex exponents.

Rationalizing the Denominator

Rationalizing the denominator is crucial when dealing with expressions that include roots or variables with negative exponents in the denominator. This technique converts such expressions into a more conventional form that does not include radicals or negative exponents in the denominator.- Negative exponents in the denominator typically imply division, meaning \(u^{-3} = \frac{1}{u^3}\). - The goal is to rewrite the expression to get rid of the negative sign when present.For our solved exercise, we end up with the expression \(2 u^{-3} v^4\) which can be rewritten as \(\frac{2v^4}{u^3}\) to avoid the negative exponent:- This step simplifies the expression by moving any terms with negative exponents from the numerator into the denominator, hence making it easier to interpret and use in further calculations.By applying rationalization, any expression becomes cleaner and more standardized, aligning with conventional mathematical practices.