Problem 56 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt{42}}{\sqrt{6}}\)

Step-by-Step Solution

Answer

\( \sqrt{7} \)

Step 1: Express as a Single Radical

Combine the radicals in the numerator and the denominator into a single radical using the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). This gives us \( \sqrt{\frac{42}{6}} \).

Step 2: Simplify the Fraction Inside the Radical

Simplify \( \frac{42}{6} \) to get \( 7 \). The expression then becomes \( \sqrt{7} \).

Step 3: Confirm Simplest Form

Verify if \( \sqrt{7} \) can be simplified further. Since 7 is a prime number, \( \sqrt{7} \) is already in its simplest radical form.

Key Concepts

Radical Expressions

Radical expressions involve roots, commonly square roots or cube roots. When you encounter a radical, like \(\sqrt{42}\), it suggests you are looking for a number which, when multiplied by itself a certain number of times, gives the original number under the root symbol.
Simply put:

• The square root \(\sqrt{n}\) equates to a number that, when squared, gives \(n\).
• The cube root \(\sqrt[3]{n}\) relates to a number that, when cubed, results in \(n\).

Understanding radicals is essential for simplifying radical expressions. You often want the simplest form because it makes further calculations easier.

Simplifying Radicals

Simplifying radicals means expressing them in their simplest form. Here, the aim is to break down the number under the radical into its prime factors, to see if pairs can be extracted.
Step-by-step for radicals:

• Express the number under the radical as a product of its prime factors.
• Identify any pairs of numbers (since we are often working with square roots, pairs are key).
• Extract each pair from under the radical as a single number outside the root.
• For cube roots, you'd look for groups of three, and so on.

In our example, \(\sqrt{42}\) divided by \(\sqrt{6}\) simplifies inside the root to \(\sqrt{7}\). Since there are no perfect squares to extract, \(\sqrt{7}\) is the simplest form.

Prime Numbers

Prime numbers are the building blocks in mathematics, crucial for simplifying radical expressions. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
For example:

• Numbers like 2, 3, 5, 7, 11, and 13 are primes.
• These numbers cannot be broken down further into a product of other numbers, which makes them key to simplifying problems.

In the context of simplifying radicals:

• If the number inside the radical is a prime, it cannot be simplified further.
• This is because there are no smaller factors to pair up and remove from under the radical.

That's why \(\sqrt{7}\) stays as it is - 7 is a prime number, indicating that \(\sqrt{7}\) is indeed in its simplest form.