Problem 62 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ \frac{1}{3} \sqrt{63} $$

Step-by-Step Solution

Answer

The simplest form of \(\frac{1}{3} \sqrt{63}\) is \(\sqrt{7}\).

Step 1: Factorize 63

The first step is to factorize the number under the square root. 63 can be expressed as the product of prime numbers: \(63 = 3 \times 3 \times 7\).

Step 2: Identify pairs and simplify

We can see that there are two 3s making a pair which can be brought out of the square root sign. This is because \(\sqrt{3 \times 3} = 3\). So, \(\sqrt{63}\) can be rewritten as \(3 \sqrt{7}\).

Step 3: Apply the fraction

The final step is to apply the fraction which was at the beginning of the expression to the simplified part. In this case, \(\frac{1}{3} \times 3 \sqrt{7}\) yields \(\sqrt{7}\).

Key Concepts

Factorization of Numbers

Factorization is the process of breaking down a number into its constituent elements, known as factors, in such a way that they can be multiplied together to give the original number. When we deal with factorization in the context of simplifying radical expressions, our goal is typically to find the prime factorization of a number.

Prime factorization involves expressing a number as a product of its prime factors, which are numbers that are only divisible by 1 and themselves. For example, when we factorize the number 63, we are looking for primes that multiply together to give us 63. In our exercise, 63 is factorized into the prime numbers 3, 3, and 7, which is written as: \(63 = 3 \times 3 \times 7\).

Understanding factorization is key because it helps us simplify radical expressions by revealing pairs of identical factors, which become crucial in the simplification process.

Square Roots

A square root of a number is a value that, when multiplied by itself, will result in the original number. The square root is represented by the radical symbol \( \sqrt{\phantom{x}} \). For example, the square root of 9 is 3 because \(3 \times 3 = 9\). When dealing with square roots in radical expressions, we often encounter numbers that aren't perfect squares.

This is where prime factorization becomes useful. If a number under a radical can be divided into a product of square numbers, those square numbers can be 'moved' outside the radical, simplifying the expression. For instance, in our textbook exercise, the number 63 isn't a perfect square, but its factorization reveals a pair of threes \(3 \times 3\), allowing us to simplify the radical.

Radical Simplification

Radical simplification is the process of reducing the complexity of a radical expression without changing its value. When simplifying a radical expression, look for any pairs of factors under the radical sign, as these can be converted into whole numbers outside the radical.

Take this exercise as an example: After factorizing 63, we found the pair of 3s, which we can extract from the radical as a single 3. The expression becomes \(3 \sqrt{7}\). When we put it all together, including the fraction from the original problem, we get \(\frac{1}{3} \times 3 \sqrt{7}\), which simplifies to \(\sqrt{7}\) because the 3 outside and the fraction \(\frac{1}{3}\) cancel each other out.

It’s essential to consistently apply all arithmetic operations as per the order of operations, mixing neither the steps of factorization nor the simplification process. Through this methodical approach, students can simplify even the most intimidating of radical expressions.