Problem 43 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{225 x^{3}}{49 x}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{15x}{7} \).

Step 1: Simplify the Fraction Inside the Radical

The given expression is \( \sqrt{\frac{225 x^{3}}{49 x}} \). Begin by simplifying the fraction inside the square root. The factors common to both the numerator and the denominator are \( x \). Divide both the numerator and the denominator by \( x \) to get \( \frac{225 x^{2}}{49} \).

Step 2: Factor the Numerator and Denominator

Factor the numerator and the denominator to make square root calculations easier. The numerator \( 225x^2 \) can be factored into \( (15x)^2 \), and the denominator 49 can be factored into \( 7^2 \). This gives \( \frac{(15x)^2}{7^2} \).

Step 3: Apply the Square Root Property

Use the property \( \sqrt{\frac{a^2}{b^2}} = \frac{a}{b} \) to simplify the square root. By applying this property in the expression \( \sqrt{\frac{(15x)^2}{7^2}} \), you get \( \frac{15x}{7} \).

Step 4: Simplified Expression

The simplified expression for \( \sqrt{\frac{225 x^{3}}{49 x}} \) is \( \frac{15x}{7} \). This is the final simplified form.

Key Concepts

Square Roots in Algebra

Square roots are interesting elements in algebra, especially when combined with fractions and variables. The square root of a number or expression is a value that, when multiplied by itself, gives the original number or expression. In algebra, we often encounter square roots that contain complex terms, such as fractions or variables, which need careful handling.

To simplify square roots involving algebraic expressions, follow these key steps:

• Identify the parts of the expression inside the square root. Pay close attention to terms that can be factored.
• Factor each term completely, which may involve variables and coefficients.
• Apply the square root property, which often involves simplifying each factor separately before dealing with the entire expression.

Understanding these steps allows you to transform a complex algebraic square root into a simpler, more manageable form. This approach streamlines solving various algebraic problems and enhances your problem-solving toolkit.

Fraction Simplification

Simplifying fractions is a crucial skill in algebra, as it often sets the stage for further simplification and solving of expressions. When dealing with algebraic fractions, which consist of variables, it’s important to reduce them to their simplest form.

To simplify a fraction:

• Identify and cancel out common factors in the numerator and the denominator. This often includes variables like \( x \) or numbers that appear in both parts.
• Look for opportunities to factor the expressions fully, which helps in finding common factors.
• Once fully factored, simplify by cancelling out the common factors. This is similar to reducing numerical fractions to their simplest form.

This crucial step can make complex equations more straightforward and easier to solve, particularly when they must interact with other operations like square roots or factorization.

Factoring in Algebra

Factoring is a foundational element in algebra, and it plays a critical role when simplifying expressions that involve square roots or fractions. By breaking down expressions into their component parts, or factors, we can simplify and solve complex algebraic problems more efficiently.

Key strategies for factoring include:

• Identifying greatest common factors (GCF), which are numbers or variables shared across terms.
• Recognizing simple patterns, like perfect square trinomials or the difference of squares, to factor quickly.
• Breaking down more complicated polynomials into simpler binomials or monomials, which can then be tackled individually.

Factoring not only aids in simplifying expressions but also in solving equations by setting each factor to zero. This technique is balanced across learning how to manage complex expressions and solve them effectively.