Problem 10 - Intermediate Algebra

Question

For Problems \(1-14\), multiply and simplify where possible. $$ (3 \sqrt{7})(2 \sqrt{7}) $$

Step-by-Step Solution

Answer

The simplified product is 42.

Step 1: Multiply the coefficients

First, identify the coefficients in the expression \((3 \sqrt{7})(2 \sqrt{7})\). The coefficients here are 3 and 2. Multiply these together: \(3 \times 2 = 6\).

Step 2: Multiply the radicands

Next, multiply the radicands: the numbers under the square roots. For this problem, both radicands are 7. So, multiply them together: \(\sqrt{7} \times \sqrt{7} = \sqrt{49}\).

Step 3: Simplify the radical

Now, simplify \(\sqrt{49}\). Since 49 is a perfect square, \(\sqrt{49} = 7\).

Step 4: Multiply the results

Finally, multiply the result from Step 1 (the coefficient) by the result from Step 3 (the simplified radical): \(6 \times 7 = 42\). So, \((3 \sqrt{7})(2 \sqrt{7}) = 42\).

Key Concepts

Multiplying Radicals

To multiply expressions that involve radicals, like square roots, you start by considering both the numerical coefficients (the numbers outside the radical symbol) and the radicands (the numbers inside the radical). In our example, where we have

• \( (3 \sqrt{7})(2 \sqrt{7}) \)

we first look at the numbers outside the radicals, known as the coefficients. These are the numbers 3 and 2 in our case. Multiply these coefficients together:\[3 \times 2 = 6\]Next, we address the radicands, which in this case are both 7. Multiply them inside the radicals like integers but keep them under the radical:\[\sqrt{7} \times \sqrt{7} = \sqrt{49}\]This is how you successfully multiply radicals.

Simplifying Expressions

Once we have multiplied the radicals, the next important step is to simplify the expression. Simplifying expressions helps in reducing radical numbers to their simplest form, making them easier to interpret and work with in further calculations. In our example, after multiplying the radicands, we end up with:\[\sqrt{49}\]The process of simplifying radicals generally involves:

• Determining if the radicand is a perfect square.
• Breaking it down to its simplest form if it is not a perfect square.

Here, since 49 is indeed a perfect square, we can simplify:\[\sqrt{49} = 7\]If it weren't a perfect square, you'd split the radicand into factors that include a perfect square and simplify separately. But in this scenario, that's unnecessary.

Perfect Squares

Understanding perfect squares is crucial when working with radicals because they allow for significant simplification of expressions. Perfect squares are numbers that can be expressed as the product of an integer with itself. For example:

• \(4 = 2 \times 2 \)
• \(9 = 3 \times 3 \)
• \(16 = 4 \times 4 \)
• \(49 = 7 \times 7 \)

So, when the radicand is a perfect square, as we saw in our example with 49, the expression simplifies neatly to its square root, i.e., \( \sqrt{49} = 7 \). Recognizing these perfect squares allows us to simplify the expression quickly, resulting in clearer and more concise calculations that can save time and reduce complexity in further math problems.