Simplify $9 \sqrt{7} \cdot \sqrt{2}-5 \sqrt{14}$

 \sqrt{7} \cdot \sqrt{2}-5 
 \sqrt{14}$

Welcome to our mathematical exploration today! We're going to tackle a problem that might look a bit intimidating at first glance, but I promise, with a few straightforward steps, we'll simplify it down to its most basic form. The expression we're working with is $9 \sqrt{7} \cdot \sqrt{2}-5 \sqrt{14}$. Our goal is to express this in its simplest form, which means combining any like terms and ensuring that any radicals we have are as simplified as possible.

Understanding the Basics of Radicals

Before we dive into the specifics of our problem, let's quickly recap some fundamental rules of radicals, also known as square roots. The most important property we'll use today is the product rule for radicals, which states that for any non-negative numbers $a$ and $b$, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This rule allows us to combine the product of two square roots into a single square root of the product of the numbers inside them. Another key concept is simplifying radicals. A radical is considered simplified if the number inside the square root (the radicand) has no perfect square factors other than 1. For example, $\sqrt{12}$ is not fully simplified because 12 has a perfect square factor of 4 ($12 = 4 \cdot 3$), so $\sqrt{12}$ can be rewritten as $\sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$. We'll apply these rules systematically to our expression.

Step 1: Simplify the Product of Radicals

Our expression begins with $9 \sqrt{7} \cdot \sqrt{2}$. Using the product rule for radicals we just discussed, we can combine $\sqrt{7} \cdot \sqrt{2}$ into a single radical: $\sqrt{7 \cdot 2}$. This simplifies to $\sqrt{14}$. So, the first part of our expression, $9 \sqrt{7} \cdot \sqrt{2}$, becomes $9\sqrt{14}$.

Now, let's substitute this back into our original expression. Our problem $9 \sqrt{7} \cdot \sqrt{2}-5 \sqrt{14}$ now looks like this: $9\sqrt{14} - 5\sqrt{14}$.

Step 2: Combine Like Radicals

We have reached a point where we have two terms, both of which contain $\sqrt{14}$. These are considered 'like radicals' because the number under the square root symbol is the same. Just like how you can combine $9x$ and $5x$ to get $4x$, you can combine $9\sqrt{14}$ and $5\sqrt{14}$. To combine like radicals, you simply add or subtract the coefficients (the numbers in front of the radical). In this case, we have $9$ and $-5$. So, we perform the subtraction: $9 - 5$. This equals $4$.

Therefore, $9\sqrt{14} - 5\sqrt{14}$ simplifies to $(9-5)\sqrt{14}$, which is $4\sqrt{14}$.

Step 3: Check for Further Simplification

The final step in simplifying any mathematical expression involving radicals is to ensure that the radical itself is in its simplest form. We have $4\sqrt{14}$. The number under the radical is 14. We need to check if 14 has any perfect square factors other than 1. The factors of 14 are 1, 2, 7, and 14. None of these (except 1) are perfect squares (like 4, 9, 16, etc.). Therefore, $\sqrt{14}$ cannot be simplified any further.

Conclusion

We started with the expression $9 \sqrt{7} \cdot \sqrt{2}-5 \sqrt{14}$, and by applying the product rule for radicals and then combining like terms, we have successfully simplified it to $4\sqrt{14}$. This is the simplest form of the original expression.

If you'd like to explore more about simplifying radicals and other algebraic manipulations, the Khan Academy website offers a wealth of resources and practice problems. They break down complex topics into easy-to-understand lessons, making it a fantastic place to deepen your mathematical understanding. For more advanced algebraic concepts, you might find the resources at MathWorld incredibly helpful, offering detailed explanations and definitions of mathematical terms and theorems.