Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt{\frac{3 x}{2 y^{3}}} $$

Step-by-Step Solution

Answer

The simplified and rationalized expression is \( \frac{\sqrt{6xy}}{2y^2} \).

Step 1: Break Down the Expression

First, let's separate the radicals in the numerator and the denominator. We have:\[ \sqrt{\frac{3x}{2y^3}} = \frac{\sqrt{3x}}{\sqrt{2y^3}} \] This simplifies the expression into two parts, each of which we can handle separately.

Step 2: Simplify the Numerator

Now, focus on simplifying the numerator \( \sqrt{3x} \). Since neither 3 nor \(x\) is a perfect square, the numerator remains as \( \sqrt{3x} \).

Step 3: Simplify the Denominator

For the denominator \( \sqrt{2y^3} \), notice that \( y^3 = y^2 \cdot y \). Then \( \sqrt{2y^3} = \sqrt{2} \times \sqrt{y^2y} = \sqrt{2} \times y \times \sqrt{y} = y\sqrt{2y} \).

Step 4: Combine and Rationalize the Denominator

Combine the simplified terms: \[ \frac{\sqrt{3x}}{y\sqrt{2y}} \] To rationalize the denominator, multiply both the numerator and denominator by \( \sqrt{2y} \): \[ \frac{\sqrt{3x}}{y\sqrt{2y}} \times \frac{\sqrt{2y}}{\sqrt{2y}} = \frac{\sqrt{3x} \cdot \sqrt{2y}}{y \cdot 2y} = \frac{\sqrt{6xy}}{2y^2} \] This gives us a rationalized expression.

Key Concepts

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form. This often means removing any perfect square factors under the square root and ensuring any expression involving a square root is reduced as much as possible. Consider our original expression \(\sqrt{\frac{3x}{2y^3}}\) and how it was broken down:

• We started by separating the expression into two parts: the numerator and the denominator.
• The aim was to simplify each part as much as possible. In our numerator, \(\sqrt{3x}\), neither 3 nor \(x\) is a perfect square, so it remains as is.
• In doing this, the task becomes simpler and more manageable.

To simplify successfully, factorize any expressions under the square root to see if any parts are perfect squares. For example, in \(\sqrt{2y^3}\), since \(y^3\) can be factorized into \(y^2 \times y\), further simplification leads to \(y\sqrt{2y}\). Breaking down radicals helps simplify algebraic expressions for easier manipulation.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations combined together. When we deal with an expression like \(\sqrt{\frac{3x}{2y^3}}\), we're working with both variables and radicals. Simplifying these expressions relies on understanding how to manipulate variables along with numbers.

• The goal is to transform the expression into a simpler form while maintaining its original value.
• Introducing steps, such as breaking down the expression into manageable parts, aids in simplification.
• Understanding properties of exponents and simplifying complex parts individually makes algebraic calculation easier.

In rationalizing, by multiplying the numerator and denominator by \(\sqrt{2y}\), variables in the expression were simplified fit for further operations. Algebra simplifies the process by which expressions are manipulated to solve problems or simplify computations in mathematics.

Square Roots

Square roots can sometimes seem tricky, especially when combined with variables. However, they are crucial in simplifying expressions and solving equations. In this expression, \(\sqrt{\frac{3x}{2y^3}}\), we had to handle square roots in both the numerator and the denominator.

• A square root is simply a value that, when multiplied by itself, gives the original number. For instance, \(\sqrt{y^2}\) equals \(y\).
• Managing square roots often involves separating the numbers into their prime factors or known squares.
  
• Once broken down, \(\sqrt{2y^3}\) became \(y\sqrt{2y}\), simplifying the expression significantly.

When rationalizing the denominator, the intention is to eliminate the square root from the denominator. This is achieved by multiplying both parts of the fraction by appropriate terms that remove the radical. Such processes make the expression easier to interpret and help in further calculations.