Problem 62 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{162}$$

Step-by-Step Solution

Answer

\(\sqrt{162}\) simplifies to \(9\sqrt{2}\).

Step 1: Prime Factorization of 162

Start by performing a prime factorization of the number 162. The prime factorization of 162 is \(2 \times 3 \times 3 \times 3 \times 3\), or more conveniently expressed as \(2 \times (3^4)\). This helps to identify the perfect squares which can be simplifed under the square root.

Step 2: Apply Square Root Properties

Next, apply square root properties to simplify the expression. The square root of a product can be expressed as the product of the square roots of the factors. So, \(\sqrt{162}\) can be rewritten as \(\sqrt{2 \times (3^4)}\), which further simplifies to \(\sqrt{2} \times \sqrt{(3^4)}\). Calculating square root of \(3^4\) gives 9 (since \(9^2 = 3^4\)). So, the simplified form would be \(9\sqrt{2}\).

Step 3: Final Simplified Form

Therefore, the final simplified form of \(\sqrt{162}\) is \(9\sqrt{2}\). This simplification process helps to reduce the complexity of the radical, making it easier to deal with in subsequent computations, if needed.

Key Concepts

Understanding Prime Factorization

Prime factorization is a fundamental step in simplifying radical expressions. It involves breaking down a number into its smallest prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, let's factorize the number 162. We start by dividing by the smallest prime number, which is 2. This gives us:

• 162 divided by 2 equals 81.
• Then, since 81 is not divisible by 2, we try the next smallest prime, which is 3.
• 81 divided by 3 equals 27.
• 27 divided by 3 equals 9.
• 9 divided by 3 equals 3.
• Finally, 3 divided by 3 equals 1.

So, the prime factorization of 162 is written as \(2 \times 3^4\). Recognizing these factors helps in the next step of simplifying the square root.

Exploring Square Root Properties

Once we have the prime factorization, understanding square root properties becomes essential. The square root of a product is the product of the square roots of the individual factors.
For our example \(\sqrt{162}\), using the prime factorization \(2 \times 3^4\), we apply this property.

• Break down \(\sqrt{162}\) into \(\sqrt{2} \times \sqrt{3^4}\).
• Notice that \(3^4\) is a perfect square \((9^2)\), so \(\sqrt{3^4} = 9\).

The simplification then becomes \(\sqrt{2} \times 9\). Knowing these properties allows you to easily simplify radicals and makes complex calculations more manageable.

Mastering Simplifying Radicals

Simplifying radicals is a strategy that revolves around making radical expressions more concise and manageable.
In our example, the task was to simplify \(\sqrt{162}\). After applying the prime factorization and square root properties, we simplified this to \(9\sqrt{2}\).

• This simplification reduces the use of large numbers and transforms the expression into an easier form to work with.
• It also facilitates additional calculations, where needed, as it removes unnecessary complexity.
• Another advantage is enhanced clarity, which is important in both mathematics and sciences where precision is vital.

Always aim for the simplest form of a radical expression to ensure efficiency in calculations.