Question

\(\sqrt{162}\) is equivalent to which of the following? A) \(3 \sqrt{2}\) B) \(9 \sqrt{2}\) C) \(54 \sqrt{3}\) D) \(2 \sqrt{81}\)

Short answer

None of the given options are equivalent to the simplified expression \(3\sqrt{6}\).

Step-by-step solution

Find the prime factors of 162

First, we need to find the prime factors of 162. Start by dividing 162 by the smallest prime number (2) and continue dividing the result until you cannot divide anymore by prime numbers. 162 ÷ 2 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 So, the prime factors of 162 are 2, 3, 3, 3.

Rewrite the square root expression using the prime factors

Now, let's rewrite the square root expression \(\sqrt{162}\) using the prime factors we found in step 1: \[\sqrt{162} = \sqrt{2 \cdot 3 \cdot 3 \cdot 3}\]

Pair the common factors

In order to simplify the square root expression, we need to pair the common factors: \[\sqrt{2 \cdot 3 \cdot 3 \cdot 3} = \sqrt{2 \cdot (3 \cdot 3) \cdot 3}\]

Simplify the square root expression

Next, simplify the square root expression by taking the square root of the paired factors: \[\sqrt{2 \cdot (3 \cdot 3) \cdot 3} = \sqrt{2} \cdot \sqrt{3 \cdot 3} \cdot \sqrt{3} = \sqrt{2} \cdot 3 \cdot \sqrt{3}\]

Calculate the final result

Lastly, multiply the simplified factors together: \[\sqrt{2} \cdot 3 \cdot \sqrt{3} = 3\sqrt{2 \cdot 3} = 3\sqrt{6}\] Comparing the simplified expression with the given options, we find that none of them are equivalent to \(3\sqrt{6}\).

Key concepts

Prime Factorization

Prime factorization is like taking a number apart to understand what makes it up. Imagine if every number was a puzzle and each piece of the puzzle was a prime number. Prime numbers, like 2, 3, 5, and 7, are numbers that can only be divided by 1 and themselves. This makes them the building blocks of all numbers. To find the prime factors of a number, you start by dividing it by the smallest prime number.

• Take 162 as an example. Begin by dividing it by 2, because 2 is the smallest prime number. 162 divided by 2 gives 81.
• Next, divide 81 by 3. Continue this process—in our case 81 divided by 3 is 27, then 27 divided by 3 is 9, and finally 9 divided by 3 is 3.
• We stop here because 3 is a prime number, and it cannot be divided any further by a smaller number besides 1.

So, the prime factors of 162 are 2 and three 3's, or as a list: 2, 3, 3, 3. Using prime factorization helps in various math problems, including simplifying square roots.

Algebra

Algebra involves using symbols, often letters, to represent numbers in equations. It's like using special placeholders in mathematical expressions. When simplifying square roots with algebra, you replace the number under the square root with its prime factors. This makes it easier to break apart and solve.
Consider the expression \( \sqrt{162} \). With prime factors, we can rewrite this as \( \sqrt{2 \cdot 3 \cdot 3 \cdot 3} \). The key to simplifying with algebra is pairing identical factors under the square root because each pair can be taken out as a single unit.

• For the square root of 162, look for pairs of the same number (3 and 3 in this case).
• You can rewrite this: \( \sqrt{2 \cdot (3 \cdot 3) \cdot 3} \).
• The square root of a pair of 3's becomes just 3, simplifying the expression to \( 3 \sqrt{2 \cdot 3} \).

This method of pairing and simplifying is a fundamental skill in algebra and helps in solving more complex algebraic problems.

Mathematics Education

Mathematics education aims to make math concepts accessible and understandable to students with various learning styles. Helping students grasp concepts like square root simplification involves breaking down processes into smaller, easier-to-understand steps. The use of step-by-step explanations and real-world examples are effective educational methods.

• For instance, teaching how to simplify \( \sqrt{162} \) can involve visual aids, like factor trees, which help students see how numbers break down into prime factors.
• Incorporating problem-solving exercises that encourage students to attempt these steps themselves enhances comprehension and retention.
• Additionally, educators often use repetitive practice to solidify knowledge—a key strategy in math education.

Connecting math concepts to everyday situations can also demystify math, turning complex algebra into something students can relate to and use practically. The goal is not only to understand and solve problems but also to develop a logical, analytical mindset for future learning.