Simplify Radicals: Multiplying Expressions With Variables

by Henrik Larsen 58 views

Hey guys! Today, we're diving into the world of multiplying radicals, specifically focusing on simplifying expressions when dealing with variables and coefficients. We'll tackle the problem: $10 \sqrt{294 v^3} \cdot 10 \sqrt{5 v^3}$, where $v$ is greater than or equal to zero. Our mission is to break down this expression step-by-step and arrive at the simplest form. So, grab your calculators and let's get started!

Understanding the Basics of Multiplying Radicals

Before we jump into the main problem, let's quickly recap the fundamental rules for multiplying radicals. Remember, the key idea is that you can multiply the coefficients (the numbers outside the square roots) and the radicands (the expressions inside the square roots) separately. So, for any non-negative numbers a, b, c, and d:

ca⋅db=c⋅da⋅bc \sqrt{a} \cdot d \sqrt{b} = c \cdot d \sqrt{a \cdot b}

This simple rule is the backbone of everything we'll be doing today. Keep it in mind as we move forward.

Breaking Down the Problem: $10 \sqrt{294 v^3} \cdot 10 \sqrt{5 v^3}$

Okay, let's get our hands dirty with the actual problem. We have $10 \sqrt{294 v^3} \cdot 10 \sqrt{5 v^3}$. The first thing we want to do is apply the rule we just discussed. We'll multiply the coefficients (10 and 10) and the radicands ($294v^3$ and $5v^3$) separately:

10⋅10294v3⋅5v310 \cdot 10 \sqrt{294 v^3 \cdot 5 v^3}

This simplifies to:

100294v3⋅5v3100 \sqrt{294 v^3 \cdot 5 v^3}

Now, let's focus on simplifying the expression inside the square root.

Simplifying the Radicand: $294 v^3 \cdot 5 v^3$

Inside the square root, we have $294 v^3 \cdot 5 v^3$. Let's multiply the numbers and the variables separately. First, we multiply the numbers:

294⋅5=1470294 \cdot 5 = 1470

Next, we multiply the variables. Remember the rule for multiplying exponents: when you multiply terms with the same base, you add the exponents:

v3⋅v3=v3+3=v6v^3 \cdot v^3 = v^{3+3} = v^6

So, our radicand simplifies to:

1470v61470 v^6

Now, let's plug this back into our expression:

1001470v6100 \sqrt{1470 v^6}

Simplifying the Square Root: $\sqrt{1470 v^6}$

Here comes the fun part: simplifying the square root. We need to break down 1470 into its prime factors and see if we can pull out any perfect squares. Let's do a prime factorization of 1470:

1470=2⋅735=2⋅3⋅245=2⋅3⋅5⋅49=2⋅3⋅5⋅7⋅7=2⋅3⋅5⋅721470 = 2 \cdot 735 = 2 \cdot 3 \cdot 245 = 2 \cdot 3 \cdot 5 \cdot 49 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 7 = 2 \cdot 3 \cdot 5 \cdot 7^2

So, we can rewrite 1470 as $2 \cdot 3 \cdot 5 \cdot 7^2$. Now, let's look at the variable part, $v^6$. Remember that $\sqrt{v^6} = v^{6/2} = v^3$. This is because taking the square root is the same as raising to the power of 1/2, and when you raise a power to a power, you multiply the exponents.

Now we rewrite the square root:

1470v6=2⋅3⋅5⋅72⋅v6\sqrt{1470 v^6} = \sqrt{2 \cdot 3 \cdot 5 \cdot 7^2 \cdot v^6}

We can take out the perfect squares: 7^2 and v^6. So, we have:

72⋅v6⋅(2⋅3⋅5)=7v32⋅3⋅5=7v330\sqrt{7^2 \cdot v^6 \cdot (2 \cdot 3 \cdot 5)} = 7v^3 \sqrt{2 \cdot 3 \cdot 5} = 7v^3 \sqrt{30}

Putting It All Together

We're almost there! Now we just need to substitute this simplified square root back into our main expression:

1001470v6=100⋅7v330100 \sqrt{1470 v^6} = 100 \cdot 7v^3 \sqrt{30}

Multiply the coefficients:

100⋅7v330=700v330100 \cdot 7v^3 \sqrt{30} = 700v^3 \sqrt{30}

And there you have it! Our final simplified expression is:

700v330700v^3 \sqrt{30}

Final Answer

So, the simplest form of $10 \sqrt{294 v^3} \cdot 10 \sqrt{5 v^3}$ is $700v^3 \sqrt{30}$. Great job, guys! We walked through each step carefully, from understanding the basic rules of multiplying radicals to simplifying the final expression. Remember, the key is to break down the problem into smaller, manageable parts and tackle each part systematically. Keep practicing, and you'll become a pro at simplifying radicals in no time!

Practice Problems for Mastering Radical Multiplication

To truly nail this concept, practice makes perfect! Here are a few more problems for you to try on your own:

  1. 518x5⋅32x35 \sqrt{18x^5} \cdot 3 \sqrt{2x^3}

  2. 275y4⋅43y22 \sqrt{75y^4} \cdot 4 \sqrt{3y^2}

  3. 932z7⋅28z39 \sqrt{32z^7} \cdot 2 \sqrt{8z^3}

Work through these problems, and don't hesitate to revisit the steps we discussed earlier. By tackling these practice problems, you'll reinforce your understanding and build confidence in simplifying radical expressions.

Common Mistakes to Avoid When Multiplying Radicals

Even though the process of multiplying radicals is straightforward, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure accurate simplifications.

  • Forgetting to Multiply Coefficients: One frequent error is only multiplying the radicands and neglecting to multiply the coefficients outside the square roots. Remember, you need to multiply both the coefficients and the radicands.
  • Incorrectly Simplifying Radicands: Make sure to break down the radicands into their prime factors correctly. A mistake in the prime factorization can lead to an incorrect simplification of the square root.
  • Adding Exponents Instead of Multiplying: When multiplying terms inside the square root, remember to add the exponents, not multiply them. For example, $v^3 \cdot v^3 = v^6$, not $v^9$.
  • Not Simplifying Completely: Always ensure that your final answer is in the simplest form. This means checking if there are any remaining perfect square factors within the radicand that can be further simplified.

By being mindful of these common mistakes, you can significantly improve your accuracy when multiplying and simplifying radicals.

Real-World Applications of Radical Multiplication

You might be wondering,