Problem 55 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt{35}}{\sqrt{7}}\)

Step-by-Step Solution

Answer

\( \sqrt{5} \)

Step 1: Express as a Single Radical

First, combine the radicals into a single expression under a single square root by using the quotient property of square roots, which says \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Apply this to get \( \frac{\sqrt{35}}{\sqrt{7}} = \sqrt{\frac{35}{7}} \).

Step 2: Simplify the Fraction Inside the Radical

Simplify the fraction inside the square root. Here, \( \frac{35}{7} \) simplifies to 5, because 35 divided by 7 equals 5. So, we have \( \sqrt{\frac{35}{7}} = \sqrt{5} \).

Step 3: Conclusion - Simplest Radial Form

Since the expression \( \sqrt{5} \) is already in its simplest form and cannot be simplified further, we've completed the transformation. Thus, \( \frac{\sqrt{35}}{\sqrt{7}} \) simplifies to \( \sqrt{5} \).

Key Concepts

Quotient Property of Square Roots

The quotient property of square roots is a very useful rule when working with radicals. It allows us to simplify expressions that involve division inside the square root. The property is expressed as:

• \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)

In simple terms, this rule tells us that the square root of the quotient of two numbers is the same as the quotient of their square roots.
This property is particularly helpful because it helps in combining two roots into one, making the expression easier to handle.
Consider this as a tool: whenever you encounter a division between square roots, you can bet on this rule to combine them into a single, more manageable radical. This first step in solving many radical expression problems simplifies the arithmetic involved in the subsequent steps.

Simplifying Fractions

Simplifying fractions is a key skill not only for working with radicals but in algebra in general. When you simplify fractions, you essentially reduce them to the lowest possible terms, ensuring you have the simplest version of the fraction.
To simplify \( \frac{35}{7} \), you need to identify the greatest common divisor (GCD) of the numbers 35 and 7.

• In this case, 7 divides both numbers perfectly.

Thus, \( \frac{35}{7} \) simplifies to 5.
This step is crucial because a simplified fraction within a radical can make the entire expression much clearer.
It reduces complexity, aids in visualization, and, as in this exercise, sometimes even results in entieredly new simpler radical forms.

Radical Expressions

Radical expressions involve roots, mainly square roots, cube roots, etc. Simplifying these expressions sometimes requires several steps, like simplifying the numbers under the root, multiplying or dividing radicals, or applying properties such as the quotient property.The ultimate goal is to bring them to their simplest form, as demonstrated in this exercise where \( \frac{\sqrt{35}}{\sqrt{7}} \) becomes \( \sqrt{5} \).

• Start with checking if the numbers can be combined using properties of radicals.
• Then, ensure any internal fractions are simplified.
• Finally, confirm the radical cannot be further broken down or simplified.

Understanding how to manipulate and simplifiy radical expressions is fundamental—it makes the process of solving equations involving roots more efficient and less error-prone. Recognize the form you're working with, apply the relevant rules, and keep practicing to master these logical steps.