Problem 47 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{\frac{x^{5} y^{3}}{z^{2}}}$$

Step-by-Step Solution

Answer

\( \frac{x^2 y \sqrt{xy}}{z} \)

Step 1: Split the Radical Expression

Start by separating the expression under the square root into different components: one for each variable. The expression is: \( \sqrt{\frac{x^{5} y^{3}}{z^{2}}} = \sqrt{x^5} \cdot \sqrt{y^3} \cdot \frac{1}{\sqrt{z^2}} \)

Step 2: Simplify Each Component

Simplify each square root individually:- For \( \sqrt{x^5} \), break it down to \( \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x} = x^2 \cdot \sqrt{x} \).- For \( \sqrt{y^3} \), break it down to \( \sqrt{y^2 \cdot y} = \sqrt{y^2} \cdot \sqrt{y} = y \cdot \sqrt{y} \).- For \( \frac{1}{\sqrt{z^2}} \), this simplifies to \( \frac{1}{z} \), since \( \sqrt{z^2} = z \).

Step 3: Combine Simplified Terms

Combine the simplified terms from the previous step:The expression becomes: \( x^2 \cdot \sqrt{x} \cdot y \cdot \sqrt{y} \cdot \frac{1}{z} = \frac{x^2 y \sqrt{xy}}{z} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves breaking down complex expressions under a square root to its simplest form. When we simplify radicals, we aim to find perfect squares or perfect 'powers' within the number or algebraic expression. This allows us to express the radical in a more manageable form.

When you have a radical expression, try to identify any factor that is a perfect square (like 4, 9, 16, or algebraically, something raised to the power of 2, 4, etc.).

• This factor can be moved outside the square root as a whole number or expression.
• For example, with \( \sqrt{50} \), realize that 50 can be split into 25 (a perfect square) and 2, as \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \).

In algebraic expressions like \( \sqrt{x^5} \), break it down to \( \sqrt{x^4 \times x} \). Since \( x^4 = (x^2)^2 \), it becomes \( x^2 \sqrt{x} \). This allows you to express the radical in a format that's easier to compute or understand in equations.

Square Roots

The concept of a square root addresses the value that, when multiplied by itself, gives the original number or expression. In algebra, managing square roots effectively is crucial, especially when simplifying them in radical expressions.

When you see \( \sqrt{y^3} \), it becomes \( \sqrt{y^2 \times y} \). Here, \( y^2 \) is a perfect square, thus can be taken out of the square root, giving \( y \sqrt{y} \). This process helps in revealing simpler and often more insightful forms of an expression.

• If you're working with variables, assume they represent positive real numbers to avoid complications with negative roots.
• The main goal is to rewrite the expression in its simplest radical form efficiently.

This practice greatly assists in comparing, adding, or multiplying radical expressions in algebraic equations or functions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (like addition, subtraction, multiplication, or division). Understanding how to manipulate these expressions is vital in algebra.

The exercise simplifies an expression involving exponents and roots, blending together various algebraic components. When working through an expression such as \( \frac{x^5 y^3}{z^2} \), the goal is to individually simplify components like \( x^5 \), \( y^3 \), and \( z^2 \), before addressing the radicals.

• Simplifying each part independently allows you to reconstruct the simplest form effectively.
• Remember, with division under a radical (like \( \sqrt{z^2} \)), simplify to the non-radical form, which makes \( \frac{1}{z} \) more straightforward.

Mastering these simplification techniques makes algebraic manipulation more intuitive and less error-prone, especially when dealing with complex equations or expressions.