Problem 66 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{60}$$

Step-by-Step Solution

Answer

The simplification of \( \sqrt{60} \) is \( 2\sqrt{15} \).

Step 1: Break into Prime Factors

Break 60 down into its prime factors. This can be done by continuously factoring the number until only prime numbers are left. The prime factorization of 60 is \(2 * 2 * 3 * 5\), so \( \sqrt{60} = \sqrt{2 * 2 * 3 * 5}\).

Step 2: Search for Pairs and Simplify

Next, search for pairs of identical prime factors. There's a single pair of 2s under the radical. We can then remove this pair from under the square root. This gives \(2\sqrt{3 * 5}\).

Step 3: Final Simplification

However, we can't simplify the expression \(2\sqrt{3 * 5}\) further as there aren't any more pairs of prime factors. So, the final result is \(2\sqrt{15}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its smallest prime factors. These are numbers that are only divisible by 1 and themselves. To find the prime factorization of a number, you will repeatedly divide it by the smallest prime numbers (2, 3, 5, etc.). Keep dividing the quotient by the smallest prime number possible until you're only left with prime numbers.

• Example: For the number 60, you divide by 2, the smallest prime number approachable from 60, resulting in 30.
• Continue by dividing 30 by 2 again to get 15.
• Since 15 is no longer divisible by 2, the next smallest prime number is 3, resulting in 5.
• Finally, 5 is already a prime number, so the process stops here.

This means 60 as a product of primes is expressed as \(2 \times 2 \times 3 \times 5\). Each of these numbers is a prime factor of 60, and this setup prepares us well for the next steps in simplifying radicals.

Simplifying Radicals

Simplifying a radical involves reducing the expression to its simplest form. The purpose is to make radical expressions easier to understand and work with. When you simplify a radical by using its prime factorization, you look for pairs of numbers under the square root symbol.

• For example, consider \(\sqrt{2 \times 2 \times 3 \times 5}\).
• The pair \(2 \times 2\) (that is, 4) is perfect because \(\sqrt{4} = 2\).
• This pair of 2s can come out from under the square root as a single 2.
• After "pulling out" the pair, you are left with \(2\sqrt{3 \times 5}\).

This process simplifies the radical by using any perfect squares contained in the original expression. If no more pairs or perfect squares remain, the expression is fully simplified, helping us move forward with mathematical tasks.

Square Roots

Square roots are a fundamental part of working with radical expressions, often represented by the radical symbol \(\sqrt{}\). The square root of a number is a value which, when multiplied by itself, gives the original number.

• For instance, the square root of 4 is 2 because \(2 \times 2 = 4\).
• When you simplify radical expressions, you're often just simplifying parts of larger equations to make solving them more manageable.
• Some numbers, such as 15 in \(2\sqrt{15}\), do not simplify further because they do not contain square numbers within their factorization.

Dealing with square roots is a common task in math, essential for simplifying expressions, solving quadratic equations, and working in geometry and algebra. Understanding square roots opens pathways to tackle complex problems efficiently.