Problem 31 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\sqrt{6} \sqrt{12}$$

Step-by-Step Solution

Answer

Question: Simplify the product of the square roots: √6 * √12 Answer: 6√2

Step 1: Recall the product rule for square roots

When multiplying square roots, we can use the product rule, which is: $$\sqrt{a} \sqrt{b} = \sqrt{ab}$$

Step 2: Apply the product rule to the given expression

Using the product rule, we can rewrite the expression as: $$\sqrt{6} \sqrt{12} = \sqrt{6 \cdot 12}$$

Step 3: Calculate the product and find the square of the prime factors

The product, 6 * 12, can be found by breaking down the factors and calculating, $$6 = 2 \cdot 3$$ $$12 = 2^2 \cdot 3$$ Then we get, $$6 \cdot 12 = 2 \cdot 2^2 \cdot 3^2 = 2^3 \cdot 3^2 = 72$$

Step 4: Simplify the square root by breaking down into perfect squares and remaining factors

To simplify the square root, we will take out the perfect square factors. In this case, both 2 and 3 have a power of 2, which is a perfect square. $$\sqrt{72} = \sqrt{2^3 \cdot 3^2}$$

Step 5: Take out the perfect squares from the square root

The perfect squares, 2^2 and 3^2 will come out of the square root as 2 and 3 respectively, leaving us with the remaining factor, 2, inside the square root. $$\sqrt{2^3 \cdot 3^2} = 2 \cdot 3 \sqrt{2}$$

Step 6: Write the final simplified expression

Multiplying the numbers outside the square root together, we get: $$2 \cdot 3 \sqrt{2} = 6\sqrt{2}$$ The simplified expression without using a calculator is, $$\sqrt{6} \sqrt{12} = 6\sqrt{2}$$

Key Concepts

Product Rule for Square Roots

The product rule for square roots is a handy tool when simplifying expressions that involve multiplying square roots. It states that if you have the square roots of two numbers, you can multiply them under a single square root. So, \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This rule helps streamline the process of simplification.
It is often used when faced with problems involving multiple square roots because it allows you to combine them first, making it easier to manage the expression. For example, in our exercise where we were simplifying \( \sqrt{6} \times \sqrt{12} \), using the product rule lets us merge these into \( \sqrt{6 \times 12} \), simplifying the rest of the operation.

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks – those numbers that multiply together to give the original number. These building blocks are called prime numbers and they are numbers greater than 1 that have no divisors other than 1 and themselves.
When simplifying radicals, prime factorization plays an essential role as it helps in identifying perfect squares, which can be taken out of the radical. For instance, the numbers 6 and 12 can be broken down as:

• 6 = 2 \( \cdot \) 3
• 12 = 2² \( \cdot \) 3

This breakdown shows the powers of numbers involved, and helps in spotting which ones can be simplified further when under a square root.

Perfect Squares

Perfect squares are fundamental when simplifying radicals because we can easily extract them from the square root. A perfect square is a number that is the square of an integer. For example, 4 (since 2² = 4) and 9 (since 3² = 9) are perfect squares.
In our exercise, after breaking down 72 as \( 2^3 \cdot 3^2 \), we recognize \( 2^2 \) and \( 3^2 \) as perfect squares. These can be 'taken out' of the square root leaving simplified terms outside of it. This means that \( \sqrt{72} \) becomes 2 and 3 outside the root, with 2 left inside, leading to \( 6\sqrt{2} \) in its simplified form. Recognizing perfect squares thus significantly reduces complexity while simplifying radicals.