Problem 33 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{2}{5} \sqrt{75}\)

Step-by-Step Solution

Answer

The simplified form is \(2\sqrt{3}\).

Step 1: Factor the Radicand

First, find the prime factorization of the number inside the radical, which is 75. We can write 75 as a product of its prime factors: \(75 = 3 imes 5^2\).

Step 2: Simplify the Radicand

Since we have \(75 = 3 \times 5^2\), we can simplify \(\sqrt{75}\) by separating the square root of the square number from the square root of the remaining number: \(\sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2} = \sqrt{3} \times 5\).

Step 3: Substitute Back into Expression

Substitute the simplified radical form back into the original expression: \(\frac{2}{5} \sqrt{75} = \frac{2}{5} \times 5 \times \sqrt{3}\).

Step 4: Simplify the Expression

Perform the multiplication: \(\frac{2}{5} \times 5 \times \sqrt{3} = 2 \times \sqrt{3}\). The \(5\) in the numerator and the \(5\) in the denominator cancel each other out, simplifying the expression further.

Key Concepts

Prime Factorization

Understanding prime factorization is crucial when simplifying radicals, especially for expressions like \(\sqrt{75}\). Prime factorization involves breaking down a number into its basic building blocks, known as prime numbers. Prime numbers are those that are greater than 1 and have no divisors other than 1 and themselves, such as 2, 3, 5, 7, 11, and so on.
To find the prime factorization of 75, you start by dividing it by the smallest prime number that can go into it evenly, which is 3.

• 75 ÷ 3 = 25
• 25 is not divisible by 3, so divide by the next smallest prime, which is 5.
• 25 ÷ 5 = 5
• And 5 ÷ 5 = 1

Combining these, you have 75 = 3 × 5 × 5, or using exponents, 75 = 3 × 5². This factorization is foundational when moving to radical simplification.

Radical Simplification

Radical simplification means reducing a radical expression to its simplest form. This form is where any square numbers under the radical are extracted or simplified out. Consider \(\sqrt{75}\) again. From the prime factorization, we know 75 = 3 × 5².

• The term 5² can come out of the radical as a single 5, since \(\sqrt{5^2} = 5\).
• Thus, \(\sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times 5\).

This process leads us to the simplest radical form of \(5\sqrt{3}\). Always aim to pull out even powers from under the root because they simplify elegantly outside the radical.

Multiplication of Radicals

When multiplying radicals, remember the basic rule: multiply the numbers outside the radicals together, and those inside the radicals with each other. In our problem, once \(\sqrt{75}\) simplifies to \(5\sqrt{3}\), we substitute it back into the original expression: \(\frac{2}{5} \times 5 \times \sqrt{3}\).

Here’s how multiplication works:

• Multiply \(\frac{2}{5}\) by 5 first, which results in just 2 (as the 5 in the numerator and denominator cancel each other out).
• You're left with 2 \(\times \sqrt{3}\).

Therefore, by multiplying factors, the original expression simplifies cleanly to \(2\sqrt{3}\). This step-by-step multiplication approach is essential in simplifying complex radical expressions smoothly.