Problem 72 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{270 y^{2}}}{5 \sqrt{3 y^{-4}}}\)

Step-by-Step Solution

Answer

The simplified expression is \( 3y^3\sqrt{2} \).

Step 1: Apply Quotient Rule

The quotient rule states that \( \frac{a}{b} = \frac{a}{b} \). For square roots, \( \frac{\sqrt{A}}{\sqrt{B}} = \sqrt{\frac{A}{B}} \). Therefore, apply this to the given expression: \( \sqrt{\frac{270y^2}{5\cdot3y^{-4}}} \).

Step 2: Simplify Inside the Square Root

Simplify inside the square root by dividing the coefficients and the variables separately: \( \frac{270}{5 \cdot 3} = \frac{270}{15} = 18 \) for the coefficients, and for the variables \( \frac{y^2}{y^{-4}} = y^{2 - (-4)} = y^6 \).

Step 3: Evaluate the Square Root

Evaluate the square root: \( \sqrt{18y^6} \). Break it down to \( \sqrt{18} \) and \( \sqrt{y^6} \). Simplify both parts. \( \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \) and \( \sqrt{y^6} = y^{3} \).

Step 4: Combine the Simplified Parts

Combine the simplified parts from the previous step: the expression becomes \( 3y^3\sqrt{2} \). This is the simplified result of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as simple as possible. We aim to reduce complex expressions to their simplest form while maintaining their equality. In the context of our exercise, this means applying mathematical rules to condense the expression

• First, we should look for like terms among coefficients and variables.
• Next, apply arithmetic operations and exponent rules to combine these like terms.
• For any complex components, use algebraic rules like the quotient rule or power rule to break them down further.

By structuring expressions this way, we uncover the simplest equivalent form. The goal of simplification is to make calculations easier and reduce chances for mistakes in further operations.

Square Roots

Square roots, represented by the symbol \( \sqrt{} \), involve understanding both the concept and operations involved. The square root of a number \( x \) is known as a number \( y \) such that \( y^2 = x \). In mathematical terms:

• The square root extracts the base of the squared number.
• It simplifies calculations involving exponents by reducing them to a more manageable form.

Within the exercise, the expression \( \frac{\sqrt{A}}{\sqrt{B}} \) transforms into \( \sqrt{\frac{A}{B}} \) due to the properties of square roots under division. This simplifies further easing of calculations by aiding in breaking down the expression into smaller parts. Simplifying square roots often involves factoring the number inside the root when possible, seeking perfect squares to extract outside—like in our exercise, where \( \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \). Understanding this approach allows deeper insight into simplifying expressions involving roots.

Variables and Exponents

Variables represent unknown values which can vary, often noted by letters such as \( x \), \( y \), etc. Exponents, on the other hand, denote repeated multiplication of base quantities. Together, they form powerful tools for mathematical expression. Let’s understand the fundamentals:

• Exponents offer a shorthand notation for expressing repeated multiplication, such as \( y^6 \) meaning \( y \times y \times y \times y \times y \times y \).
• When dividing variables with exponents, subtract the exponent of the divisor from the exponent of the dividend, following: \( \frac{y^m}{y^n} = y^{m-n} \).

In this exercise, the rule \( \frac{y^2}{y^{-4}} = y^{2 - (-4)} = y^6 \) illustrates how handling negative exponents correctly leads to multiplication rather than division. Simplifying expressions with variables and exponents not only shortens them but also simplifies subsequent calculations and reveals underlying mathematical relationships.