Problem 1 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{5} \sqrt{6} $$

Step-by-Step Solution

Answer

Answer: The product of √5 and √6 is √30.

Step 1: Write the given expression

We are given an expression as follows: $$ \sqrt{5} \cdot \sqrt{6} $$

Step 2: Multiply the square roots

We'll use the rule of multiplying square roots: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). In this case, \(a = 5\) and \(b = 6\): $$ \sqrt{5} \cdot \sqrt{6} = \sqrt{5 \cdot 6} $$

Step 3: Simplify the product inside the square root

Next, we will simplify the product inside the square root by multiplying 5 and 6: $$ \sqrt{5 \cdot 6} = \sqrt{30} $$

Step 4: Final answer

Therefore, the product of the given square roots is: $$ \sqrt{5} \cdot \sqrt{6} = \sqrt{30} $$

Key Concepts

Radical Multiplication

Multiplying square roots, also known as radical multiplication, follows a specific rule that simplifies the process. This rule is key to handling square roots when they are part of a multiplication problem.
Consider two square roots, \(\sqrt{a}\) and \(\sqrt{b}\). The multiplication of these radicals can be rearranged by multiplying the numbers under the radicals directly, which means \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). This is a fundamental concept because it allows us to focus on the values beneath the radical signs rather than each radical individually.

• Ensures a straightforward way to deal with square roots in expressions.
• Works even if the square roots contain fractions or decimals.

By applying this rule, you reduce complexity and make it easier to work with these expressions.

Simplifying Radicals

Once you have multiplied the radicands, the next step is simplifying them, if possible. Simplifying radicals involves expressing the number inside the square root in its simplest form. If the product under the square root \(\sqrt{ab}\) can be broken down to include perfect squares, you can further simplify it.
For instance, if after multiplication, you have \(\sqrt{36}\), you know that 36 is a perfect square (since \(6\times6=36\)), and thus it simplifies to 6.

• Find factors of the number inside the radical.
• Check for perfect squares and simplify accordingly.

However, if no perfect square factors exist, like \(\sqrt{30}\), the expression is already in its simplest form. Not all square roots can be further simplified, and recognizing this is crucial to avoid unnecessary steps.

Product of Square Roots

The concept of the product of square roots is straightforward when you apply the rule for multiplying radicals. The key is to ensure that you multiply correctly and check if the result can be simplified.
In the original exercise, you multiplied \(\sqrt{5}\) and \(\sqrt{6}\) to get \(\sqrt{30}\). Here, there are no perfect square factors within 30, indicating the product is in its simplest form.

• Combining square roots should lead directly to a simpler radical if possible.
• Always verify the factors to ensure the final radical simplifies completely.

Ultimately, mastering the product of square roots enables you to efficiently handle many mathematical problems involving radicals, streamlining calculations and enhancing accuracy.