Problem 79 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt{\frac{16 u^{3} v}{u v^{5}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4uv^{-2}\).

Step 1: Simplify Inside the Square Root

Start by simplifying the expression inside the square root: \[ \frac{16 u^{3} v}{u v^{5}} \]Divide \(u^{3}\) by \(u\) to get \(u^{2}\). Similarly, divide \(v\) by \(v^{5}\) to get \(v^{-4}\). Hence, the expression simplifies to: \[ 16 u^{2} v^{-4} \]

Step 2: Apply the Square Root to Each Component

Apply the square root to each component separately: \[ \sqrt{16 u^{2} v^{-4}} = \sqrt{16} \times \sqrt{u^{2}} \times \sqrt{v^{-4}} \]Calculate each square root:- \(\sqrt{16} = 4\)- \(\sqrt{u^{2}} = u\)- \(\sqrt{v^{-4}} = v^{-2}\)

Step 3: Combine the Simplified Components

Combine the simplified components from the previous step:\[ 4u v^{-2} \]

Step 4: Express the Final Answer Using Rational Exponents

Rewrite the expression \(4u v^{-2}\) using rational exponents:\[ 4u v^{-2} = 4u v^{-2} \]This is already in terms of rational exponents, as \(v^{-2}\) is equivalent to \(\frac{1}{v^{2}}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about breaking down complex mathematical expressions into simpler, more manageable forms. This process often involves combining like terms, reducing fractions, and eliminating unnecessary components. By simplifying expressions, you make it easier to perform further calculations or analyze the expression.

• Identify like terms and combine them to reduce the complexity.
• Factorize when possible to simplify fractions or polynomial expressions.
• Ensure that every term is expressed in its simplest form by using the basic rules of algebra.

In the expression from our original exercise, simplification involved reducing the quotient \( \frac{u^3 v}{u v^5} \) to \( 16 u^2 v^{-4} \). This was done by dividing the terms with similar bases. Simplifying in this way reduces the difficulty of handling the expression in future steps.

Square Roots

Understanding square roots is essential for simplifying expressions involving powers and exponents. A square root represents a number that, when multiplied by itself, gives the original number. In expressions like \(\sqrt{16 u^2 v^{-4}}\), you need to apply the square root to each term independently.

• The square root of a perfect square, such as 16, is straightforward: \(\sqrt{16} = 4\).
• Variables with even powers simplify neatly: for instance, \(\sqrt{u^2} = u\).
• For negative exponents within square roots, like \(v^{-4}\), the square root operation halves the exponent, giving \( v^{-2} \).

By breaking them into individual components, you can systematically tackle each part of the expression and merge them back once simplified.

Exponents and Powers

Rational exponents and powers help express roots and repeated multiplications compactly. An exponent denotes how many times we multiply a base by itself. Rational exponents offer a precise way to express roots as powers, which simplifies mathematical operations.

• Rational exponents like \( v^{-2} = \frac{1}{v^2} \) provide clarity by converting roots to fractional powers.
• When dealing with exponents, use the rule: \( a^{m/n} = \sqrt[n]{a^m} \). This way, square roots (\(\sqrt{\cdot}\)) are expressed as \((\cdot)^{1/2}\).
• Negative exponents signify reciprocals: \( v^{-n} = \frac{1}{v^n} \).

In the original problem, the expression \(4u v^{-2}\) uses rational exponents to indicate that \(v\) should be squared and reciprocated. This shifts the expression from an implicit root to an explicitly defined exponent form, showcasing a key benefit of using rational exponents.