Problem 27 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{12 x^{3} y^{3}}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 2xy\sqrt{3xy} \).

Step 1: Prime Factorization of Coefficients

First, factor the numerical part under the square root: 12 can be factored into prime numbers as \( 12 = 2^2 \times 3 \).

Step 2: Simplify Variables' Exponents

The expression under the square root is \( x^3 \times y^3 \). Since we are finding square roots, we look for pairs in the exponents. Therefore, \( x^3 = x^2 \times x \) and \( y^3 = y^2 \times y \).

Step 3: Extract Square Root Components

Now, under the square root we have \( \sqrt{2^2 \times 3 \times x^2 \times x \times y^2 \times y} \). We can now take \( 2 \), \( x \), and \( y \) out of the square root because they are perfect squares: \( \sqrt{2^2} = 2 \), \( \sqrt{x^2} = x \), and \( \sqrt{y^2} = y \).

Step 4: Write the Expression with Remaining Terms

After taking out the perfect squares, we are left with \( 2xy\sqrt{3xy} \). Only the terms without complete pairs (\(3xy\)) stay under the square root.

Key Concepts

Prime Factorization

Prime factorization is the first step in simplifying radicals, particularly when you have a numerical coefficient under the square root. We aim to break down that number into its basic prime factors, which are numbers that can only be divided by 1 and themselves. Here's how it works:

• Take the number 12 for instance. Our goal is to express 12 as a product of prime numbers.
• Start by dividing 12 by the smallest prime number, which is 2. We get: 12 ÷ 2 = 6
• Divide 6 by 2 again: 6 ÷ 2 = 3
• 3 is a prime number, so we stop here.

Thus, we have expressed 12 as a product of its prime factors: \(12 = 2^2 \times 3\).
This factorization allows us to extract the square root of those parts that form a perfect square, which is essential to simplifying radicals.

Square Root Extraction

Once we have broken down the number into prime factors, the next task is to extract perfect squares from under the radical. Perfect squares are numbers like 4, 9, 16, etc., that result from squaring a whole number.

• For example, in this exercise, we find that 12 is factored as \(2^2 \times 3\).
• The term \(2^2\) is a perfect square because \((2 imes 2) = 4\).
• Thus, the square root of \(2^2\) can be simplified to 2.

Doing this helps simplify the radicals by removing the perfect square components from inside the square root. What remains in the square root are those factors that don't have a pair. Here, \(3\) remains inside the radical because it does not form a perfect square with any other factor. This process gives us the simplified form \(2\sqrt{3}\) for the numerical part.

Variables' Exponents

When dealing with variables under a square root, the process of simplification focuses on their exponents. This involves finding and forming pairs since taking the square root of something squared simplifies it.

• Consider the expression \(x^3 \times y^3\).
• We can express \(x^3\) as \(x^2 \times x\) and \(y^3\) as \(y^2 \times y\).
• Both \(x^2\) and \(y^2\) are perfect squares since \(\sqrt{x^2} = x\) and \(\sqrt{y^2} = y\).

Therefore, we extract \(x\) and \(y\) out of the square root. After extraction, the remaining expressions under the radical don’t entirely square up and thus stay inside the square root. So the simplified term becomes \(xy\sqrt{xy}\).

• This technique helps in systematically simplifying expressions with variables under square roots.