Problem 77 - Algebra 1

Question

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

Step-by-Step Solution

Answer

\(\sqrt{60}\) simplifies to \(2\sqrt{15}\).

Step 1: Find Prime Factors

Start by finding the prime factors of 60. It can be broken down into two factors as follows \[60 = 6 * 10\]. Both these can be further broken down: \[6 = 2 * 3\] and \[10 = 2 * 5\]. Thus, the prime factorization of 60 is \(2 * 2 * 3 * 5\). Both of the 2's are a pair, which makes them a perfect square.

Step 2: Take out the Perfect Square

Take this perfect square (i.e., \(2*2\)), square root it and take it out from under the square root symbol. This leaves the other factors (i.e., 3 and 5) which multiply to 15, under the square root. The square root of \(2*2\) is 2. So now, the expression can be written as \[2\sqrt{15}\]. This cannot be simplified further as 15 does not have any perfect square factors.

Step 3: Write Final Answer

So \(\sqrt{60}\) simplifies to \[2\sqrt{15}\]. This is the final simplified form of the radical expression.

Key Concepts

Prime Factorization

When simplifying a radical expression, such as \(\sqrt{60}\), one of the first steps is finding its prime factorization. Prime factorization involves breaking down a number into its smallest unit factors, which are prime numbers. Prime numbers are numbers that can only be divided by 1 and themselves.

For the number 60, begin by choosing any two numbers that multiply to 60, such as 6 and 10.

• For 6, its prime factors are 2 and 3, since \(6 = 2 \times 3\).
• For 10, the prime factors are 2 and 5, because \(10 = 2 \times 5\).

Therefore, the complete prime factorization of 60 is \(2 \times 2 \times 3 \times 5\). Identifying the prime factors is crucial for the next steps in simplifying radicals.

Radical Expression

A radical expression is any expression that has a square root, cube root, or any higher root symbol involved, like \(\sqrt{60}\). Simplifying radical expressions means reducing them to their simplest form while keeping them exact under the root symbol as much as possible. This involves incorporating prime factors to extract perfect squares, if any, out from under the root.

For \(\sqrt{60}\), we use the prime factorization \(2 \times 2 \times 3 \times 5\). We look for pairs of numbers, because square roots involve pairs (i.e., two identical numbers that multiply together). In this case, the pair is the two 2s. This pair can be taken out of the radical as a single 2, since \(\sqrt{2 \times 2} = 2\).
After extracting 2 from the radical, the remaining product inside the square root is \(3 \times 5 = 15\). Hence the expression \(\sqrt{60}\) simplifies to \(2\sqrt{15}\), since 15 doesn’t contain any pairs and remains inside the square root. Reading radical expressions in this form is more concise and often required in both homework and tests.

Perfect Squares

Perfect squares play a significant role in simplifying radicals, as they readily emerge from the prime factorization process. A perfect square is any number that can be expressed as some integer squared, like 1, 4, 9, 16, and so on.

When dealing with \(\sqrt{60}\), finding the perfect square factor within its prime factorization allowed us to simplify the expression. Identifying the pair of 2s allowed us to recognize the perfect square of 4 (because \(2 \times 2 = 4\)), which simplified our radical. Simplifying means pulling the square root of this perfect square out as its integer equivalent, which in the case of our problem was 2.

• Ensure that in each expression, perfect squares are moved outside the radical.
• Leave non-perfect square factors inside to maintain accuracy.

Recognizing perfect squares promptly helps streamline the simplification of any radical expression, such as \(2\sqrt{15}\), which shows perfect use of knowing how to handle these numbers in expressions.