Problem 25 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \sqrt{\frac{3}{7}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$\frac{\sqrt{3}}{\sqrt{7}}$$.

Step 1: Identify components of fraction

We are given the expression $$\sqrt{\frac{3}{7}}$$. We can see that the numerator is 3 and the denominator is 7.

Step 2: Calculate the square root of numerator and denominator separately

We will now simplify the numerator and the denominator individually: Square root of 3 (numerator): \(\sqrt{3}\) Square root of 7 (denominator): \(\sqrt{7}\)

Step 3: Combine square roots into simplified expression

As the square root of a fraction can be written as the ratio of the square roots of the numerator and the denominator, we can combine them to get the simplified expression: $$\sqrt{\frac{3}{7}} = \frac{\sqrt{3}}{\sqrt{7}}$$

Step 4: Final Simplification

In its current form, the expression is already simplified. Hence, the final simplified expression is: $$\boxed{\frac{\sqrt{3}}{\sqrt{7}}}$$

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots—where the square root is one of the most common types. A square root, denoted by the radical symbol \( \sqrt{ } \), essentially asks the question, 'What number multiplied by itself gives you the original number?' For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \. When dealing with rational numbers (fractions), as in the exercise \( \sqrt{\frac{3}{7}} \), we can look at the numerator and the denominator separately because \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). The rules for simplifying radical expressions indicate that we should simplify each part independently, just as the solution process shows.

The Art of Square Root Simplification

Simplifying a square root involves expressing the number under the radical as simple as possible. Sometimes this means finding perfect squares within the number to simplify. For example, \( \sqrt{8} \) can be broken down into \( \sqrt{4 \times 2} \) which simplifies to \( 2\sqrt{2} \). However, in our case where we have \( \sqrt{\frac{3}{7}} \), neither 3 nor 7 is a perfect square. Consequently, we simplify by separately addressing the square roots of the numerator and denominator, resulting in \( \frac{\sqrt{3}}{\sqrt{7}} \). This expression is in its simplest radical form as neither \( \sqrt{3} \) nor \( \sqrt{7} \) can be simplified further.

Fractional Exponents as an Alternate Form

Fractional exponents are another way to represent powers and roots. The expression \( a^{\frac{1}{n}} \) is equivalent to the nth root of 'a' (\( \sqrt[n]{a} \) or \( a^{1/n} \) in the context of square roots). Specifically, a square root such as \( \sqrt{a} \) can also be written as \( a^{\frac{1}{2}} \). When faced with a fractional expression under a square root like \( \sqrt{\frac{3}{7}} \), one could rewrite this as \( \left(\frac{3}{7}\right)^{\frac{1}{2}} \). This format can sometimes make it easier to work with expressions involving powers and can also illuminate properties like \( \left(a^{m}\right)^{n} = a^{m \cdot n} \), though in this simplified case, it doesn't change the function of the expression.