Problem 344 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{15 x^{3}}}{\sqrt{3 x}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x \sqrt{5} \).

Step 1 - Combine the Radicals

Using the property of radicals, \ \( \frac{ \sqrt{a} }{ \sqrt{b} } = \sqrt{ \frac{a}{b} } \), write the expression as a single radical: \ \[ \frac{ \sqrt{15 x^{3}} }{ \sqrt{3 x} } = \sqrt{ \frac{15 x^{3}}{3 x} } \]

Step 2 - Simplify the Expression Inside the Radical

Simplify the fraction inside the radical: \ \[ \frac{15 x^{3}}{3 x} \] \ Divide both the numerator and the denominator by 3x: \ \[ \frac{15 x^{3}}{3 x} = \frac{15}{3} \cdot \frac{x^{3}}{x} = 5 \cdot x^{2} \]

Step 3 - Simplify the Radical

Taking the square root of the simplified expression: \ \[ \sqrt{5 x^{2}} \] \ Use the property of radicals, \ \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \): \ \[ \sqrt{5 x^{2}} = \sqrt{5} \cdot \sqrt{x^{2}} \] \ Since \( \sqrt{x^{2}} = x \), the expression simplifies to: \ \[ x \sqrt{5} \]

Key Concepts

properties of radicals

Radicals deal with roots, such as square roots and cube roots. To simplify radical expressions, we use certain properties. One useful property is that the division of two radicals can be combined into a single radical. For example, if you have \( \frac{ \sqrt{a} }{ \sqrt{b} } \), it can be rewritten as \[ \sqrt{ \frac{a}{b} } \]. This property helps to simplify expressions by reducing the number of radicals. Another important property is that of multiplying radicals, such as \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). It means that the product under a radical can be split into separate radicals. Understanding these properties can make handling complex expressions easier and more intuitive.

simplification of fractions

Simplifying fractions is about reducing the numerator and the denominator to their smallest values. For example, in \( \frac{15 x^{3}}{3 x} \), we can simplify by dividing both the numerator and the denominator by the greatest common factor, which in this case is 3x. So, \[ \frac{15 x^{3}}{3 x} = \frac{15}{3} \cdot \frac{x^{3}}{x} = 5 \cdot x^{2} \]. By breaking down each part into its simplest form, simplifying fractions helps make further calculations and interpretations easier. It’s an essential skill in algebra, ensuring clarity and accuracy in more complex problems.

algebraic expressions

An algebraic expression can include numbers, variables, and operations like addition, subtraction, multiplication, and division. In simplifying algebraic expressions, especially those with radicals, you apply various algebraic rules and properties. For instance, to simplify \[ \sqrt{5 x^{2}} \], you can split it using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \), leading to \[ \sqrt{5 x^{2}} = \sqrt{5} \cdot \sqrt{x^{2}} = \sqrt{5} \cdot x = x \sqrt{5} \]. Knowing how to manipulate and simplify these expressions is fundamental in algebra and makes solving equations much simpler. It’s like translating a complex statement into something easily understandable.