Problem 70 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{5}{b}} $$

Step-by-Step Solution

Answer

Question: Simplify the given radical expression: $$\sqrt{\frac{5}{b}}$$ Answer: The simplified expression is: $$\frac{\sqrt{5}}{\sqrt{b}}$$

Step 1: Identify the given expression

The given expression is: $$ \sqrt{\frac{5}{b}} $$

Step 2: Simplify the radical expression

Recall that, for a fraction, we can take the square root of the numerator and the square root of the denominator separately, and then divide them. So we will simplify the given expression by taking the square root of both the numerator and the denominator: $$ \sqrt{\frac{5}{b}} = \frac{\sqrt{5}}{\sqrt{b}} $$

Step 3 (Final): Write the simplified expression

The simplified expression is: $$ \frac{\sqrt{5}}{\sqrt{b}} $$

Key Concepts

Radical Expression Simplification

When dealing with radical expressions, the goal is to simplify the expression so that it’s easier to understand and work with. Simplification often involves processes such as factoring, dividing the radical into simpler parts, or rationalizing the denominator. Simplifying does not change the value of an expression, it simply makes it cleaner and more straightforward.

For instance, the radical \(\sqrt{ab}\) could be expressed as \(\sqrt{a} \cdot \sqrt{b}\) if both 'a' and 'b' are non-negative. The process should always be executed by keeping an eye on the mathematical rules, such as the product rule of radicals or the rule that roots of fractions can be separated into the root of the numerator and the root of the denominator respectively.

Square Root of Fractions

When taking the square root of fractions, it is important to handle the numerator and the denominator separately. This can be done because of a handy rule of square roots that states \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) when 'a' and 'b' are positive real numbers and 'b' is not zero.

This rule is particularly useful since it transforms the initial complex fraction into an easier form where we can deal with the square roots of the numerator and the denominator independently. For example, the square root of the fraction \(\frac{5}{b}\) is split into \(\frac{\sqrt{5}}{\sqrt{b}}\). However, if 'b' were to be a perfect square (like 4 or 9), its square root would further simplify to an integer and the expression would become even simpler.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They are the building blocks of algebra and are used to represent real-world and mathematical problems. Simplifying algebraic expressions is critical for solving equations efficiently.

In our exercise, the algebraic expression under the radical sign involved variables and fractions. In algebra, fractions are often present and treating them correctly is vital for simplification. Always consider the domain of the variable to avoid division by zero or taking square roots of negative numbers in a real number system context, as these actions are undefined. Understanding how to manipulate algebraic expressions with square roots can greatly assist in solving more complex equations involving radicals.