Problem 2 - Intermediate Algebra

Question

For Problems \(1-14\), multiply and simplify where possible. $$ \sqrt{8} \sqrt{6} $$

Step-by-Step Solution

Answer

\(4\sqrt{3}\)

Step 1: Multiply the Radicals

Multiply the numbers under the square roots: \[ \sqrt{8} \times \sqrt{6} = \sqrt{48} \]

Step 2: Simplify the Radical

Simplify \(\sqrt{48}\) by finding the largest perfect square factor of 48. Notice that \(48 = 16 \times 3\), where 16 is a perfect square. Thus, \[\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

Key Concepts

Multiplying Radicals

When it comes to **multiplying radicals**, the process is simpler than it may first appear. The key is to multiply the numbers under the square root symbols directly. For example, if you have \( \sqrt{a} \times \sqrt{b} \), you can combine them into one square root: \( \sqrt{a \times b} \). Using the exercise, \( \sqrt{8} \times \sqrt{6} \) becomes \( \sqrt{48} \). This multiplication does not change individual values; it simply combines them under a single radical.

• In our example, we multiplied 8 and 6 to get 48.
• The key rule is that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).

Multiplying radicals helps consolidate multiple square roots into a more manageable expression under a single radical sign. This sets the stage for the simplification process in the next steps.

Perfect Square Factorization

**Perfect square factorization** is a technique used to simplify radicals further. It involves finding a factor of a number that is a perfect square. A perfect square is a number that is the square of an integer, like 4, 9, 16, 25, etc.
When you look at \( \sqrt{48} \), the goal is to express 48 as a product of factors, one of which is a perfect square. For 48, you can break it down into 16 and 3 because 16 is a perfect square. So, \( 48 = 16 \times 3 \), enabling you to rethink \( \sqrt{48} \) as \( \sqrt{16 \times 3} \).

• Identify the largest perfect square factor of the number under the radical.
• Express the number as a product of that perfect square and another factor.

Recognizing perfect squares in numbers allows you to take those terms out of the radical, simplifying your expression significantly.

Simplifying Square Roots

**Simplifying square roots** involves reducing the expression to its simplest form. Once you've identified a perfect square factor, you can proceed to simplify the radicals. From the breakdown of \( 48 = 16 \times 3 \), you rewrite \( \sqrt{48} \) as \( \sqrt{16} \times \sqrt{3} \). Since \( \sqrt{16} = 4 \) (because 16 is a perfect square), the expression simplifies to \( 4\sqrt{3} \).

• Extract the square root of the perfect square factor.
• Multiply this outside the radical by any remaining terms inside the radical.

In doing so, the radical expression is reduced to \( 4\sqrt{3} \), which is simpler and more manageable than \( \sqrt{48} \). Simplifying square roots makes it easier to work with radicals in mathematical problems. It helps you recognize value and symmetry within more complex algebraic expressions.