Simplifying Radicals: Multiplying $\sqrt{7x^5} \cdot \sqrt{7x}$

Alright, let's dive into simplifying some radicals! Today, we're tackling the expression $\sqrt{7x^5} \cdot \sqrt{7x}$. This might look a bit intimidating at first, but don't worry, we'll break it down step by step to make it super easy to understand. So, grab your pencil, and let's get started!

Understanding the Basics of Radical Multiplication

Before we jump directly into the problem, let's quickly review the basics of multiplying radicals. Remember that when you multiply two square roots, you can combine them under a single square root. In other words: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This rule is super important and will guide us through the simplification process. Also, keep in mind that we're looking for pairs of factors inside the square root because each pair can be pulled out as a single factor outside the square root. This is because $\sqrt{x^2} = x$. Understanding these fundamental concepts will make simplifying any radical expression a breeze!

When simplifying radicals, especially those involving variables, it’s crucial to remember the properties of exponents. For instance, when multiplying terms with the same base, you add their exponents: $x^m \cdot x^n = x^{m+n}$. This property is extremely useful when dealing with expressions like $x^5 \cdot x$, which simplifies to $x^6$. Moreover, knowing how to break down exponents into pairs is essential for pulling terms out of the square root. For example, $x^6$ can be seen as $(x^3)^2$, allowing us to simplify $\sqrt{x^6}$ to $x^3$. These exponent rules, combined with the basic principles of radical multiplication, form the backbone of simplifying more complex expressions. It’s also worth noting that ensuring the variables are non-negative is a common assumption in these types of problems, which allows us to avoid complications with absolute values. Therefore, mastering these basics will not only help in this specific problem but also in a wide range of algebraic manipulations. Remember, practice makes perfect, so keep applying these rules to various problems to solidify your understanding!

Always remember to check for common factors that can be simplified before combining the terms under the radical. This preliminary step can often make the subsequent calculations easier and reduce the risk of errors. Also, be mindful of the index of the radical (in this case, it's a square root). If you were dealing with cube roots or higher-order radicals, the approach would need to be adjusted accordingly, looking for groups of three or more identical factors instead of pairs. Make sure to understand the question fully before you get started with the calculations. It's also a good idea to rewrite the question on your notebook.

Step-by-Step Solution

Now, let's apply this to our problem: $\sqrt{7x^5} \cdot \sqrt{7x}$.

1. Combine the Radicals: Using the rule $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, we can combine the two square roots into one: $\sqrt{7x^5 \cdot 7x}$.
2. Simplify Inside the Radical: Multiply the terms inside the square root: $7 \cdot 7 = 49$ and $x^5 \cdot x = x^6$. So we have $\sqrt{49x^6}$.
3. Simplify the Square Root: Now, we simplify $\sqrt{49x^6}$. We know that $\sqrt{49} = 7$ and $\sqrt{x^6} = x^3$ (since $x^6 = (x^3)^2$).
4. Final Answer: Putting it all together, we get $7x^3$.

And that's it! The simplified form of $\sqrt{7x^5} \cdot \sqrt{7x}$ is $7x^3$.

Detailed Breakdown of Each Step

Let's break down each step in detail to make sure we've got a solid understanding.

Combining the Radicals

The initial step involves using the property of radicals that allows us to combine two separate square roots into a single square root when they are being multiplied. This property, expressed as $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, is a fundamental rule in simplifying radical expressions. Applying this rule to our problem, $\sqrt{7x^5} \cdot \sqrt{7x}$, we combine the contents of the two radicals under one square root, resulting in $\sqrt{7x^5 \cdot 7x}$. This step simplifies the expression into a form where we can perform further simplifications by combining like terms and identifying perfect squares.

Simplifying Inside the Radical

Once the terms are under a single radical, the next step is to simplify the expression inside the square root. This involves multiplying the constants and combining the variables using the properties of exponents. In our case, we have $\sqrt{7x^5 \cdot 7x}$. Multiplying the constants, $7 \cdot 7$ gives us $49$. Multiplying the variables, $x^5 \cdot x$, we use the rule $x^m \cdot x^n = x^{m+n}$, which means $x^5 \cdot x^1 = x^{5+1} = x^6$. Therefore, the expression inside the square root simplifies to $49x^6$. This simplification makes it easier to recognize and extract perfect squares from the radical.

Simplifying the Square Root

After simplifying the expression inside the square root, we move on to extracting the square root of the simplified expression. Here, we have $\sqrt{49x^6}$. We know that $\sqrt{49}$ is $7$ because $7 \cdot 7 = 49$. For the variable part, we need to find the square root of $x^6$. Since $x^6$ can be written as $(x^3)^2$, the square root of $x^6$ is $x^3$. Thus, $\sqrt{x^6} = x^3$. Combining these, we get $\sqrt{49x^6} = 7x^3$. This step essentially reverses the squaring operation and simplifies the radical expression to its simplest form.

Final Answer

Finally, after simplifying each part of the square root, we combine the simplified terms to obtain the final answer. From the previous steps, we found that $\sqrt{49} = 7$ and $\sqrt{x^6} = x^3$. Combining these, we get $7x^3$. Therefore, the simplified form of the original expression $\sqrt{7x^5} \cdot \sqrt{7x}$ is $7x^3$. This is our final simplified expression, which is much easier to work with than the original.

Practice Problems

To solidify your understanding, here are a few practice problems:

1. $\sqrt{3x^2} \cdot \sqrt{3x^4}$
2. $\sqrt{5a^3} \cdot \sqrt{5a^7}$
3. $\sqrt{2y^6} \cdot \sqrt{8y^2}$

Try these out and see if you can simplify them correctly! The more you practice, the easier it will become. Remember to follow the steps we discussed: combine the radicals, simplify inside, and then simplify the square root.

Conclusion

So, there you have it! Multiplying and simplifying radicals might seem tricky at first, but with a solid understanding of the basic rules and a bit of practice, you'll be simplifying like a pro in no time. Just remember to combine the radicals, simplify the expression inside, and then take the square root. Keep practicing, and you'll master these types of problems effortlessly. Keep up the great work, and remember that every step you take in understanding mathematics brings you closer to mastering it. Good luck, and happy simplifying!