Solving Radical Expressions: A Step-by-Step Guide

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Solving Radical Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of radical expressions. If you've ever felt a little intimidated by those square root symbols, don't worry! We're going to break down two problems step-by-step, so you'll be a pro in no time. We'll tackle two problems: $5 \sqrt{2} \cdot -2 \sqrt{20}$ and $3 \sqrt{18 w^7} \cdot 10 \sqrt{4 w^9}$. Let’s get started and demystify these radical equations together!

Problem 1: $5 \sqrt{2} \cdot -2 \sqrt{20}$

Let's start with our first expression: $5 \sqrt{2} \cdot -2 \sqrt{20}$. This might look a bit daunting at first, but we can simplify it using some basic rules of radicals. The key here is to remember that we can multiply the coefficients (the numbers outside the square root) and the radicands (the numbers inside the square root) separately. Our focus here is on clearly explaining each step, making it super easy to follow along and grasp the core concepts. So, stay with me as we break down each part of this radical equation!

Step 1: Multiply the Coefficients

First, let’s multiply the coefficients, which are the numbers outside the square root. In this case, we have 5 and -2. Multiplying these together is pretty straightforward:

5imes−2=−105 imes -2 = -10

So, the coefficient part of our answer is -10. Remember, when dealing with multiplication, a positive number times a negative number results in a negative number. This simple rule is crucial and will guide you in solving similar problems. It sets the stage for the rest of the calculation, so make sure you’ve got this down!

Step 2: Multiply the Radicands

Next, we’ll multiply the radicands, which are the numbers inside the square root symbols. Here, we have 2{ \sqrt{2} } and 20{ \sqrt{20} }. When we multiply these, we get:

2imes20=2imes20=40\sqrt{2} imes \sqrt{20} = \sqrt{2 imes 20} = \sqrt{40}

So, now we have 40{ \sqrt{40} }. Remember, the product of the radicands goes under a single square root. This step helps us combine the square roots into a single term, making it easier to simplify further. Keep this in mind as it’s a fundamental part of handling radicals.

Step 3: Simplify the Radical

Now, let's simplify 40{ \sqrt{40} }. To do this, we need to find the largest perfect square that divides 40. Perfect squares are numbers like 4, 9, 16, 25, etc., which are the squares of whole numbers. In this case, the largest perfect square that divides 40 is 4. We can rewrite 40{ \sqrt{40} } as:

40=4imes10\sqrt{40} = \sqrt{4 imes 10}

Now we can separate the square root:

4imes10=4imes10\sqrt{4 imes 10} = \sqrt{4} imes \sqrt{10}

Since 4=2{ \sqrt{4} = 2 }, we have:

2102 \sqrt{10}

This simplification is key because it breaks down the complex radical into simpler terms. We look for perfect square factors to make the simplification process smoother and more manageable. Identifying these factors is a critical skill in dealing with radicals.

Step 4: Combine the Results

Finally, we combine the results from Step 1 and Step 3. We found that the product of the coefficients is -10 and the simplified radical is 210{ 2 \sqrt{10} }. Multiplying these together gives us:

−10imes210=−2010-10 imes 2 \sqrt{10} = -20 \sqrt{10}

So, the simplified form of $5 \sqrt{2} \cdot -2 \sqrt{20}$ is $-20 \sqrt{10}$. Yay, we did it! Combining these simplified components is the final touch, bringing everything together into a clean and understandable result. This step demonstrates the power of breaking down a problem into manageable parts and then reassembling them.

Problem 2: $3 \sqrt{18 w^7} ext{*} 10 \sqrt{4 w^9}$

Alright, let's move on to our second problem: $3 \sqrt{18 w^7} \cdot 10 \sqrt{4 w^9}$. This one looks a little more complex with the variables and exponents, but don't worry! We'll tackle it using the same principles we used before. The inclusion of variables and exponents adds an extra layer, but it also provides an opportunity to see how these concepts integrate with radical simplification. Our goal is to make this process as clear and straightforward as possible.

Step 1: Multiply the Coefficients

Just like before, let's start by multiplying the coefficients. In this case, we have 3 and 10:

3imes10=303 imes 10 = 30

So, our coefficient is 30. Nice and easy, right? Multiplying the coefficients is always a great first step because it simplifies the overall expression right from the start. It’s a consistent approach that works well across different radical problems.

Step 2: Multiply the Radicands

Now, let's multiply the expressions inside the square roots:

18w7imes4w9=18w7imes4w9\sqrt{18 w^7} imes \sqrt{4 w^9} = \sqrt{18 w^7 imes 4 w^9}

Multiplying the numbers and variables inside the radical, we get:

18imes4imesw7imesw9=72w7+9=72w16\sqrt{18 imes 4 imes w^7 imes w^9} = \sqrt{72 w^{7+9}} = \sqrt{72 w^{16}}

Remember the rule for multiplying exponents with the same base: you add the exponents. So, w7×w9=w16{ w^7 \times w^9 = w^{16} }. This step showcases the importance of exponent rules in simplifying radicals. Combining these rules effectively is key to handling more complex expressions.

Step 3: Simplify the Radical

Next, we simplify 72w16{ \sqrt{72 w^{16}} }. Let's break this down. First, we simplify the number part, 72. The largest perfect square that divides 72 is 36 (since 36imes2=72{ 36 imes 2 = 72 }). So, we can rewrite 72{ \sqrt{72} } as:

72=36imes2=36imes2=62\sqrt{72} = \sqrt{36 imes 2} = \sqrt{36} imes \sqrt{2} = 6 \sqrt{2}

Now, let's simplify the variable part, w16{ w^{16} }. Since 16 is an even number, we can take the square root of w16{ w^{16} } by dividing the exponent by 2:

w16=w16/2=w8\sqrt{w^{16}} = w^{16/2} = w^8

So, 72w16{ \sqrt{72 w^{16}} } simplifies to 62imesw8=6w82{ 6 \sqrt{2} imes w^8 = 6 w^8 \sqrt{2} }. Breaking down the radical into numerical and variable parts makes the simplification process much easier. It allows us to apply different simplification techniques to each part individually.

Step 4: Combine the Results

Finally, we combine the results from Step 1 and Step 3. We have the coefficient 30 and the simplified radical expression 6w82{ 6 w^8 \sqrt{2} }. Multiplying these together gives us:

30imes6w82=180w8230 imes 6 w^8 \sqrt{2} = 180 w^8 \sqrt{2}

Therefore, the simplified form of $3 \sqrt{18 w^7} \cdot 10 \sqrt{4 w^9}$ is $180 w^8 \sqrt{2}$. Awesome! Combining the coefficient and the simplified radical expression gives us the final, neat answer. This last step ties everything together, showing how each simplification contributes to the overall solution.

Final Thoughts on Simplifying Radicals

And there you have it, guys! We've successfully simplified two radical expressions. Remember, the key is to break the problem down into smaller, manageable steps. First, multiply the coefficients, then multiply the radicands, simplify the radical by finding perfect square factors, and finally, combine your results. Simplifying radicals might seem tricky at first, but with practice, it becomes second nature. Breaking down complex problems into manageable steps is a skill that extends far beyond mathematics. Keep practicing, and you'll be simplifying radicals like a pro in no time! Keep up the great work, and I’ll see you in the next mathematical adventure!