Simplifying Radicals Multiply 4√(10t⁵) ⋅ 8√(154t)

Hey guys! Today, we're diving deep into the world of radical expressions, specifically how to multiply them when variables are involved. We'll be tackling a problem where we need to simplify the expression $4 sqrt{10 t^5} \cdot 8 \sqrt{154 t}$, assuming that $t \ge 0$. This is a common type of problem in algebra, and mastering it will definitely boost your math skills. So, grab your pencils, and let's get started!

Breaking Down the Problem

Understanding the Basics

Before we jump into the solution, let's quickly review what radicals are and how they work. A radical is a mathematical expression that involves a root, such as a square root, cube root, or any nth root. The most common type is the square root, denoted by the symbol $\sqrt{}$. The number inside the radical is called the radicand. When we multiply radicals, there are a few key rules we need to keep in mind. First, we can only multiply radicals that have the same index. The index is the small number that sits in the crook of the radical symbol (e.g., the '2' in a square root, although it's usually not written). If the indices are the same, we can multiply the radicands together and keep the same index. Also, remember that the product rule for radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, where a and b are non-negative. This rule is the cornerstone of multiplying and simplifying radicals.

Initial Assessment of the Expression

Now, let's look at our expression: $4 \sqrt{10 t^5} \cdot 8 \sqrt{154 t}$. We have two radical terms being multiplied together. Notice that both radicals are square roots (index of 2), so we can proceed with multiplying them. The expression also involves variables, specifically $t$, which adds a bit of complexity but nothing we can't handle. Our goal is to simplify this expression as much as possible. This means we want to remove any perfect square factors from the radicand and write the expression in its most compact form. We also need to keep in mind the condition that $t \ge 0$, which ensures that we're dealing with real numbers and avoids any issues with taking the square root of a negative number.

The Importance of Simplest Form

Simplifying radicals is not just a mathematical exercise; it's about presenting the answer in the most understandable and usable form. A simplified radical has no perfect square factors left inside the radical, and the radicand is as small as possible. Think of it like reducing a fraction to its lowest terms. Just as $\frac{2}{4}$ is technically correct but not in its simplest form (which is $\frac{1}{2}$), a radical expression can be correct but not fully simplified. Simplifying radicals makes them easier to compare, combine, and use in further calculations. It's a crucial skill for algebra and beyond. For example, in physics or engineering, you might encounter radical expressions when dealing with distances, areas, or volumes. Simplifying these expressions can make calculations much easier and less prone to error.

Step-by-Step Solution

Step 1: Multiply the Coefficients and Radicands

The first step in simplifying $4 \sqrt10 t^5} \cdot 8 \sqrt{154 t}$ is to multiply the coefficients (the numbers outside the radicals) and the radicands (the expressions inside the radicals) separately. Multiply the coefficients $4 \cdot 8 = 32$. Multiply the radicands: $10 t^5 \cdot 154 t = 1540 t^6$. So, our expression now becomes $32 \sqrt{1540 t^6$. This step combines the two separate radical terms into a single term, making it easier to work with. It's like combining like terms in an algebraic expression; we're grouping the numbers and variables together to simplify the overall expression. This multiplication step is a direct application of the product rule for radicals, which allows us to bring the radicands under a single radical sign as long as the indices are the same. Remember, this rule only works for multiplication (and division); we can't use it for addition or subtraction of radicals unless they have the exact same radicand.

Step 2: Prime Factorization of the Radicand

Next, we need to prime factorize the radicand, 1540$t^6$ This means breaking down 1540 into its prime factors. Prime factorization is essential for identifying any perfect square factors within the radicand. Start by dividing 1540 by the smallest prime number, 2: $1540 = 2 \cdot 770$. Continue dividing by 2 until you can't anymore: $770 = 2 \cdot 385$. Now, 385 is not divisible by 2, so try the next prime number, 3. It's not divisible by 3 either, so try 5: $385 = 5 \cdot 77$. Finally, $77 = 7 \cdot 11$. So, the prime factorization of 1540 is $2^2 \cdot 5 \cdot 7 \cdot 11$. For the variable part, $t^6$, we can rewrite it as $(t³⁾2$, which is a perfect square. Thus, the prime factorization of the entire radicand $1540t^6$ is $2^2 \cdot 5 \cdot 7 \cdot 11 \cdot (t³⁾2$. This step is crucial because it helps us identify the perfect square factors that we can take out of the radical. Prime factorization is a fundamental tool in number theory and is used in various areas of mathematics, including cryptography and computer science. It's a technique that allows us to understand the building blocks of a number and simplify complex expressions.

Step 3: Simplify the Radical

Now, we can simplify the radical by taking out the perfect square factors. From the prime factorization $2^2 \cdot 5 \cdot 7 \cdot 11 \cdot (t³⁾2$, we can see that $2^2$ and $(t³⁾2$ are perfect squares. We can take the square root of these factors and move them outside the radical. $\sqrt2^2} = 2$ and $\sqrt{(t³⁾2} = t^3$. So, we bring 2 and $t^3$ outside the radical, multiplying them with the coefficient 32 $32 \cdot 2 \cdot t^3 = 64 t^3$. The remaining factors inside the radical are 5, 7, and 11. Multiply them together: $5 \cdot 7 \cdot 11 = 385$. Thus, our simplified expression becomes $64 t^3 \sqrt{385$. This step is the heart of simplifying radicals. By identifying and extracting perfect square factors, we reduce the radicand to its smallest possible value. This not only makes the expression more manageable but also reveals the true nature of the radical. Simplifying radicals is like cleaning up a messy room; you're taking out the clutter and organizing the important stuff. In mathematics, this leads to clearer and more efficient calculations.

Final Answer

Therefore, the simplified form of $4 \sqrt{10 t^5} \cdot 8 \sqrt{154 t}$ is $64 t^3 \sqrt{385}$. We multiplied the coefficients, combined the radicands, performed prime factorization, and extracted the perfect square factors. This methodical approach ensures that we've simplified the expression completely. Remember, the key to mastering radical expressions is practice. The more problems you solve, the more comfortable you'll become with the process. So, keep practicing, and you'll be a radical pro in no time!

Common Mistakes and How to Avoid Them

Forgetting to Multiply Coefficients

A common mistake is to only multiply the radicands and forget to multiply the coefficients outside the radical. Always remember to multiply the numbers outside the radicals together as well. This is a simple mistake, but it can lead to a completely wrong answer. To avoid this, make it a habit to explicitly write down the multiplication of the coefficients as a separate step. For example, in our problem, we first multiplied 4 and 8 to get 32 before dealing with the radicals.

Incorrect Prime Factorization

Another frequent error is making mistakes during prime factorization. It's crucial to break down the radicand correctly to identify all perfect square factors. Double-check your work and ensure you've listed all the prime factors. A helpful tip is to use a factor tree or division ladder to systematically break down the number. This visual approach can make it easier to keep track of the factors and avoid missing any. Also, remember to include the variable part of the radicand in your prime factorization. For example, we had to factorize $t^6$ as $(t³⁾2$ to identify the perfect square.

Missing Perfect Square Factors

Sometimes, you might correctly factorize the radicand but miss a perfect square factor. This can happen if you're not careful or if the numbers are large. Make sure to look for pairs of identical factors, as these indicate perfect squares. A good practice is to circle or highlight the pairs of prime factors that form perfect squares. This will make it easier to see which factors can be taken out of the radical. In our example, we identified $2^2$ and $(t³⁾2$ as perfect squares and extracted them from the radical.

Not Simplifying Completely

A common pitfall is to simplify the radical partially but not completely. This means you might take out some perfect square factors but leave others behind. Always double-check the radicand after each simplification step to ensure there are no more perfect square factors left. If you're unsure, try dividing the radicand by the smallest perfect squares (4, 9, 16, etc.) to see if there are any factors that can be further simplified. Remember, the goal is to reduce the radicand to its smallest possible value.

Ignoring the Non-Negative Condition

In problems where variables are involved, pay attention to any conditions given, such as $t \ge 0$. This condition is crucial because it ensures that we're dealing with real numbers and avoids taking the square root of a negative number. In our example, the condition $t \ge 0$ allowed us to directly take the square root of $t^6$ as $t^3$. If the condition were different, we might need to use absolute value signs to ensure the result is non-negative.

Practice Problems

To solidify your understanding, let's try a few more practice problems:

1. $$3 \sqrt{8 x^3} \cdot 5 \sqrt{6 x}$$

2. $$2 \sqrt{27 y^7} \cdot 4 \sqrt{12 y}$$

3. $$\sqrt{20 a^4 b^3} \cdot \sqrt{35 a b^2}$$

Work through these problems step-by-step, following the same procedure we used in the example. Remember to multiply coefficients, combine radicands, prime factorize, and extract perfect square factors. Don't forget to double-check your work and simplify completely. The more you practice, the more confident you'll become in multiplying and simplifying radicals. You've got this!

Conclusion

Multiplying radicals with variables might seem daunting at first, but with a systematic approach and a bit of practice, it becomes a manageable task. Remember the key steps: multiply the coefficients and radicands, prime factorize the radicand, and extract the perfect square factors. By avoiding common mistakes and working through practice problems, you'll master this skill and be well-prepared for more advanced algebra concepts. Keep up the great work, and happy simplifying!