Problem 11 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{60}\) is \(2*\sqrt{15}\).

Step 1: Prime Factorization of the Number

Prime factorize the number 60. This can be done by dividing 60 by smallest prime number (2), successively until the result is a prime number. The prime factors we get are \(2, 2, 3, 5\). So, we can rewrite \(\sqrt{60}\) as \(\sqrt{2*2*3*5}\).

Step 2: Simplify the Square Root

In this step, identify pairs of same numbers, as the square root of a pair of the same numbers is that number. Our pair is \(2*2\). So, we move this pair of 2's out from the square root sign. Now our expression becomes \(2*\sqrt{3*5}\).

Step 3: Compute the Result

Now, you simply need to write out the remaining part of the expression which is under the square root sign. So, the simplified expression becomes \(2*\sqrt{15}\).

Key Concepts

Prime Factorization

Prime factorization is a crucial initial step in simplifying square roots. It involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than itself and 1. To find the prime factors of a number like 60, follow these simple steps:

• Start dividing the number by the smallest prime number, which is usually 2.
• Continue dividing the result by 2 until you cannot divide evenly anymore.
• Once you finish with 2, move to the next smallest prime number, which is 3, and so on.

This process ultimately gives us the factorization of 60 as 2, 2, 3, and 5. Writing it in a multiplied form, we get 60 = 2 × 2 × 3 × 5. Prime factorization simplifies the process of dealing with radicals by allowing us to identify easily which numbers can come out of the root.

Radical Expression

A radical expression involves a root symbol, like a square root (√). When dealing with a square root, simplifying the expression makes it easier to understand and calculate. The square root of a number finds a value that, when multiplied by itself, gives the original number.

In the context of simplifying radicals, once we have a number expressed in terms of its prime factors, we can simplify the radical expression. For instance, with \[ \sqrt{60} = \sqrt{2 \times 2 \times 3 \times 5} \] our goal is to juggle these prime numbers to simplify the square root by identifying pairs. Each pair of identical numbers under the square root can be simplified by taking the number outside the root, as it represents being squared inside. In this scenario, the pair of 2s can be taken out as a single 2, leading us to a simplified expression: \[ 2 \sqrt{15} \] which is now less cumbersome and much easier to manipulate.

Square Root Properties

Understanding the properties of square roots is essential to simplifying expressions involving radicals. Two fundamental properties will help us:

• First, the square root of a product is the product of the square roots: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
• Second, the power rule, which tells us the square root of a number squared is the absolute value of the original number: \[ \sqrt{x^2} = |x| \]

In our example of simplifying \[ \sqrt{60} \] we apply the first property by breaking down 60 into its prime factors and rewriting it beneath the square root. Then, we use the square of a particular pair from the factorization (2 in our case), applying the second property to bring a 2 outside the square root. These properties empower us to manipulate and simplify square roots efficiently, offering a strategic approach to mastering radicals.