Problem 69 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{2}{a}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: $\sqrt{\frac{2}{a}}$ Answer: $\frac{\sqrt{2}}{\sqrt{a}}$

Step 1: Rewrite the expression as a product of square roots

In the given expression, we have a fraction inside the square root: $$ \sqrt{\frac{2}{a}} $$ According to the property of square roots, we can rewrite this as the product of two square roots: $$ \sqrt{\frac{2}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} $$

Step 2: Simplify the square root

Now we will simplify the square root of the fraction: $$ \sqrt{\frac{1}{a}} $$ This can be rewritten using the same property from step 1 as: $$ \sqrt{\frac{1}{a}} = \frac{\sqrt{1}}{\sqrt{a}} = \frac{1}{\sqrt{a}} $$

Step 3: Write the final simplified expression

We will now use steps 1 and 2 to write the simplified expression: $$ \sqrt{\frac{2}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \frac{1}{\sqrt{a}} = \frac{\sqrt{2}}{\sqrt{a}} $$ Hence, the simplified radical expression is: $$ \frac{\sqrt{2}}{\sqrt{a}} $$

Key Concepts

Square Roots in Radical Expressions

Understanding square roots is crucial when simplifying radical expressions. A square root, often written as \( \sqrt{x} \), is essentially asking what number multiplies by itself to give the number \( x \). For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).
When you encounter a radical expression like \( \sqrt{\frac{2}{a}} \), it involves finding the square root of a fraction. This can initially seem complex. However, there's a rule that can simplify it: You can break the square root of a fraction into the product of two separate square roots. That means \( \sqrt{\frac{2}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} \).
Separating square roots makes the process more straightforward and clearer. Once broken down, it's easier to address each component separately, simplifying the overall expression.

Simplifying Fractions Involving Square Roots

When dealing with fractions inside a square root, breaking it down helps simplify it. Let's consider the part of the expression \( \sqrt{\frac{1}{a}} \).
Here, it's split into \( \frac{\sqrt{1}}{\sqrt{a}} \). This is because \( \sqrt{\frac{1}{a}} \) can be seen as a division of two individual square roots.
- \( \sqrt{1} \) is simply 1 because any number multiplied by itself gives 1.- \( \sqrt{a} \) is left as is because typically, \( a \) is not a perfect square.
After simplifying, this part of the expression becomes \( \frac{1}{\sqrt{a}} \). Understanding this step is vital for mastering fraction simplification in algebra.

Algebraic Expressions with Radicals

Algebraic expressions often include variables and exponents, and understanding these elements is important when working with radicals.
Algebraic expressions with radicals like \( \sqrt{2} \cdot \frac{1}{\sqrt{a}} \) involve both constants and variables. Applying the properties of radicals to such expressions is essential for simplification.
By understanding how each component—either a number or a letter—interacts within the radical, we can simplify these complex expressions step-by-step. For instance:- \( \sqrt{2} \) stays the same because it cannot be reduced further.- \( \frac{1}{\sqrt{a}} \) is how the expression changes, as previously simplified in the previous section.
The skill is to practice recognizing patterns and applying consistent rules, such as separating radicals across fractions, to achieve simplified forms like \( \frac{\sqrt{2}}{\sqrt{a}} \). This insight leads to a more manageable understanding of more intricate algebraic manipulations.