Simplifying Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem: simplifying the expression $2ab\left(\sqrt[3]{192ab^2}\right) - 5\left(\sqrt[3]{81a^4b^5}\right)$. This might look a bit intimidating at first, but trust me, we'll break it down step-by-step and make it super easy to understand. We're going to use our knowledge of radicals, prime factorization, and some basic algebra to get a simplified answer. So, grab your pencils, and let's get started!

Breaking Down the First Term: $2ab\left(\sqrt[3]{192ab^2}\right)$

Our first step is to focus on the first term: $2ab\left(\sqrt[3]{192ab^2}\right)$. This part involves a cube root, so we need to look for perfect cubes hidden inside the radical. Let's start with the number 192. We need to find the prime factorization of 192. This means breaking it down into a product of prime numbers. Let's do it:

• $192 = 2 \times 96$
• $96 = 2 \times 48$
• $48 = 2 \times 24$
• $24 = 2 \times 12$
• $12 = 2 \times 6$
• $6 = 2 \times 3$

So, the prime factorization of 192 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3$, or $2^6 \times 3$. Now we can rewrite the first term as: $2ab\left(\sqrt[3]{2^6 \times 3 \times a \times b^2}\right)$.

Next, let's use the properties of cube roots. A cube root of a product is the product of the cube roots. Since we're dealing with a cube root, we're looking for groups of three identical factors. We have $2^6$, which is the same as $(2^3 \times 2^3)$. So, we can take out a $2^2$ (which is 4) from the cube root. The term becomes: $2ab \times 2^2 \sqrt[3]{3ab^2}$. Simplifying further, we get $2ab \times 4 \sqrt[3]{3ab^2}$. Thus, the first term simplifies to $8ab\sqrt[3]{3ab^2}$.

To make this crystal clear, we're essentially taking the cube root of any perfect cubes inside the radical. For example, the cube root of $2^6$ (which is 64) is 4, because $4 \times 4 \times 4 = 64$. We pulled out the factors that could be simplified and left the remaining factors inside the cube root. We did the same thing with the variables. We will simplify the second term in a similar manner.

Simplifying the Second Term: $5\left(\sqrt[3]{81a^4b^5}\right)$

Alright, let's tackle the second term: $5\left(\sqrt[3]{81a^4b^5}\right)$. First things first, let's find the prime factorization of 81. It's $3 \times 3 \times 3 \times 3$, or $3^4$. So, we can rewrite the second term as: $5\left(\sqrt[3]{3^4 \times a^4 \times b^5}\right)$.

Now, let's break down the variables inside the cube root. For the $a^4$, we can rewrite it as $a^3 \times a$. For $b^5$, we can rewrite it as $b^3 \times b^2$. Now our term looks like this: $5\left(\sqrt[3]{3^4 \times a^3 \times a \times b^3 \times b^2}\right)$.

Time to pull out those perfect cubes! We have $3^4$ (which is $3^3 \times 3$), $a^3$, and $b^3$. Taking the cube roots, we get $3$, $a$, and $b$. So, we have: $5 \times 3ab\sqrt[3]{3ab^2}$. Simplifying, we get $15ab\sqrt[3]{3ab^2}$.

Remember, we are looking for groups of three because we're dealing with a cube root. So, we took out a 3 from $3^4$ (leaving one 3 inside), one 'a' from $a^4$ (leaving one 'a' inside), and one 'b' from $b^5$ (leaving two 'b's inside). Always break it down into as many groups of three as possible, then simplify. This process might seem tedious at first, but with practice, it becomes much easier!

Combining the Simplified Terms

Okay, we've simplified both terms! Let's put them back together. Remember, the original problem was $2ab\left(\sqrt[3]{192ab^2}\right) - 5\left(\sqrt[3]{81a^4b^5}\right)$. We found that:

• The first term simplifies to $8ab\sqrt[3]{3ab^2}$.
• The second term simplifies to $15ab\sqrt[3]{3ab^2}$.

So, our problem now looks like this: $8ab\sqrt[3]{3ab^2} - 15ab\sqrt[3]{3ab^2}$. Notice something cool? Both terms have the same radical part: $\sqrt[3]{3ab^2}$. This means we can combine them like terms, just like we would with $8x - 15x$.

So, $8ab\sqrt[3]{3ab^2} - 15ab\sqrt[3]{3ab^2} = (8 - 15)ab\sqrt[3]{3ab^2}$. And $8 - 15 = -7$. Therefore, our final answer is $-7ab\sqrt[3]{3ab^2}$.

That's it, guys! We successfully simplified the expression. See, it wasn't so bad after all! We used prime factorization, properties of radicals, and basic algebra to get our answer. Remember to take it step-by-step, and always look for those perfect cubes inside the radicals.

Key Concepts Recap

Let's quickly recap the key concepts we used:

• Prime Factorization: Breaking down numbers into a product of prime numbers.
• Cube Roots: Finding a number that, when multiplied by itself three times, equals the original number.
• Properties of Radicals: The cube root of a product is the product of the cube roots. For example, $\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}$.
• Combining Like Terms: Adding or subtracting terms that have the same variable and exponent (or, in this case, the same radical).

Practice Makes Perfect!

The best way to get good at simplifying radical expressions is to practice! Try working through similar problems on your own. Start with simpler ones and gradually work your way up to more complex expressions. Don't be afraid to make mistakes – that's how we learn. The more you practice, the more comfortable and confident you'll become. You'll soon be simplifying radicals like a pro. Keep up the fantastic work, and you'll do great. Practice, and you'll become a master of radicals in no time! Keep practicing, and you'll be acing those math problems in no time. Good luck, and happy simplifying!