Problem 26 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{y^{2} z}{225}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{y \cdot \sqrt{z}}{15} \).

Step 1: Introduce the Problem

We need to simplify the expression \( \sqrt{\frac{y^{2} z}{225}} \) using the quotient rule. Remember that the quotient rule for square roots states that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Step 2: Apply the Quotient Rule

Apply the quotient rule to the expression:\[ \sqrt{\frac{y^{2} z}{225}} = \frac{\sqrt{y^{2} z}}{\sqrt{225}} \].

Step 3: Simplify the Denominator

Determine the square root of the denominator:\( \sqrt{225} = 15 \). Thus, we have:\[ \frac{\sqrt{y^{2} z}}{15} \].

Step 4: Simplify the Numerator

Simplify \( \sqrt{y^{2} z} \). Use the property \( \sqrt{a b} = \sqrt{a} \cdot \sqrt{b} \):\[ \sqrt{y^{2} z} = \sqrt{y^{2}} \cdot \sqrt{z} = y \cdot \sqrt{z} \].

Step 5: Combine Results

Substitute \( \sqrt{y^{2} z} \) back into the expression:\[ \frac{\sqrt{y^{2} z}}{15} = \frac{y \cdot \sqrt{z}}{15} \].

Step 6: Final Answer

The simplified form of the original expression is:\[ \frac{y \cdot \sqrt{z}}{15} \].

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions is crucial to making complex equations more manageable. A radical expression involves roots, and our goal is to simplify these terms to their most reduced form. Consider the expression \( \sqrt{\frac{y^{2} z}{225}} \). Here, each component under the square root can potentially be simplified individually. By breaking down the radical further, we can express the terms in their simplest forms, thereby making calculations and further algebraic manipulations easier.

• Identify which operations can be applied to different parts of the expression.
• Use properties such as distributing the square root over multiplication or division.
• Simplify each part of the expression while ensuring all steps are mathematically valid.

Ultimately, the aim is to express the radical in the simplest possible way, such as writing \( \sqrt{y^2} \) as \( y \) when \( y \) is a positive real number.

Square Roots

Square roots are one type of radical where the index is 2. It involves finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 225 is 15, because \( 15 \times 15 = 225 \). When dealing with expressions involving square roots, like \( \sqrt{\frac{y^2 z}{225}} \), knowing this allows us to separate and simplify these terms.In the context of the quotient rule for square roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) can be applied. This rule helps break down complex square roots into more manageable components:

• First, apply the rule to simplify the denominator, \( 225 \), by finding its square root directly.
• For the numerator, \( y^2 \) simplifies, as \( \sqrt{y^2} = y \), making the expression easier to handle.

Square roots simplify further calculations, reducing a multi-variable equation into simpler, easily interpretable components.

Properties of Exponents

Understanding the properties of exponents is essential when simplifying expressions involving variables raised to powers, such as \( y^2 \). These properties allow us to manipulate and simplify expressions efficiently. For example, one of the key properties is that \( (a^m)^n = a^{m \cdot n} \), which can be used to break down exponents or extract common terms.When simplifying radicals like \( \sqrt{y^2} \), we utilize the property that \( \sqrt{a^2} = a \) for positive values, which simplifies \( y^2 \) to \( y \). This basic understanding helps in handling more complicated expressions:

• Identify terms that are perfect squares to simplify square roots effectively.
• Combine like terms using properties of exponents to ease further simplification.

Ultimately, properties of exponents are instrumental in transforming complex algebraic expressions into simpler forms. This greatly aids in solving equations and understanding the underlying mathematical concepts.