Problem 10 - Intermediate Algebra

Question

Use the product rule to multiply. See Example \(I\). $$ \sqrt{\frac{6}{m}} \cdot \sqrt{\frac{n}{5}} $$

Step-by-Step Solution

Answer

\(\sqrt{\frac{6n}{5m}}\)

Step 1: Apply the Product Rule for Radicals

The product rule for radicals states that for any two non-negative real numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Apply this rule to \(\sqrt{\frac{6}{m}} \cdot \sqrt{\frac{n}{5}}\), resulting in \(\sqrt{\frac{6}{m} \cdot \frac{n}{5}}\).

Step 2: Multiply the Fractions

To multiply \(\frac{6}{m}\) by \(\frac{n}{5}\), multiply the numerators together and the denominators together: \(\frac{6 \cdot n}{m \cdot 5} = \frac{6n}{5m}\).

Step 3: Simplify Within the Radical

Substitute \(\frac{6n}{5m}\) back into the radical expression to get \(\sqrt{\frac{6n}{5m}}\). This is the final simplified expression.

Key Concepts

Multiplying Radicals

Multiplying radicals might seem tricky at first, but once you understand the product rule, it becomes much simpler. The product rule for radicals tells us that when you multiply two square roots together, you're allowed to combine them under a single square root sign. This rule works as long as the numbers under the square roots, called the radicands, are non-negative.In practical terms, this means if you have two radicals, like \( \sqrt{a} \cdot \sqrt{b} \), you can rewrite it as \( \sqrt{a \cdot b} \). In our exercise, this translates to taking the expression \( \sqrt{\frac{6}{m}} \cdot \sqrt{\frac{n}{5}} \) and combining it into \( \sqrt{\frac{6}{m} \cdot \frac{n}{5}} \). This step simplifies multiplication of radicals significantly, allowing you to deal with just one radical instead of multiple, scattered ones.

Simplifying Radical Expressions

Once you've applied the product rule, the next step is simplifying the expression inside the radical. Simplifying is about making the expression as neat and straightforward as possible.For our exercise, we've combined the fractions under the radical to get \( \sqrt{\frac{6n}{5m}} \). At this point, check if anything inside this radical can be simplified further. Typically, you should look for common factors in the numerator and denominator or perfect squares within the expression. If there were any, you could simplify them to make the expression tidier. For this problem, however, we find that \( \frac{6n}{5m} \) is already in its simplest form.Remember, simplifying radicals is all about patience and attention to detail. Go step by step, checking if further simplification is possible to make the expression as clean and readable as possible.

Fractions Multiplication

Multiplying fractions is an essential part of dealing with radical expressions like the one in this exercise. The rule is straightforward: multiply the numerators (the top parts) together and do the same for the denominators (the bottom parts).In our exercise, we need to multiply \( \frac{6}{m} \) by \( \frac{n}{5} \). So, for the numerators, you multiply 6 by \( n \), and for the denominators, \( m \) by 5, resulting in \( \frac{6n}{5m} \).Ensure you always simplify both the numerators and denominators separately if possible, before multiplying them together. While the final step here didn't require simplification beyond multiplication, many other problems might, so always keep an eye out for any common factors. Understanding this process is key to mastering fractions multiplication within radical expressions.