Reduced Radical Form: $\sqrt[3]{27x^8 Y^{19}}$

Let's break down how to express $\sqrt[3]{27x^8 y^{19}}$ in its simplest radical form. This involves understanding the properties of radicals and exponents, and then applying them strategically to simplify the given expression.

Understanding Radicals and Exponents

Before we dive into the specific problem, let's quickly recap some fundamental concepts. A radical is simply a root, like a square root, cube root, etc. The general form is $\sqrt[n]{a}$, where n is the index (the type of root) and a is the radicand (the value inside the root). Exponents, on the other hand, represent repeated multiplication. They're closely related to radicals, as a radical can be expressed as a fractional exponent. For example, $\sqrt[n]{a} = a^{\frac{1}{n}}$.

The key here is recognizing perfect cubes (since we have a cube root). Perfect cubes are numbers or expressions that can be obtained by cubing another number or expression. For instance, 8 is a perfect cube because $2^3 = 8$. Similarly, $x^6$ is a perfect cube because $(x^2)^3 = x^6$. Recognizing these perfect cubes within the radicand is crucial for simplification.

Remember these exponent rules:

• $(a^m)^n = a^{mn}$ (Power of a power)
• $a^m \cdot a^n = a^{m+n}$ (Product of powers)
• $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ (Product under a radical)

With these rules in mind, we can tackle simplifying the given expression.

Simplifying $\sqrt[3]{27x^8 y^{19}}$

Now, let's simplify $\sqrt[3]{27x^8 y^{19}}$ step by step.

1. Factor the Radicand: First, we want to break down the radicand ($27x^8 y^{19}$) into its prime factors and identify any perfect cubes. We know that 27 is $3^3$, which is a perfect cube. For the variables, we need to express the exponents as multiples of 3 plus a remainder.

   • $27 = 3^3$
   • $x^8 = x^{6+2} = x^6 \cdot x^2 = (x^2)^3 \cdot x^2$
   • $y^{19} = y^{18+1} = y^{18} \cdot y^1 = (y^6)^3 \cdot y$

   So, we can rewrite the expression as:

   $\sqrt[3]{3^3 \cdot x^6 \cdot x^2 \cdot y^{18} \cdot y}$

2. Separate the Perfect Cubes: Next, we separate out the perfect cubes from the rest of the radicand:

   $\sqrt[3]{3^3 x^6 y^{18} \cdot x^2 y} = \sqrt[3]{3^3 x^6 y^{18}} \cdot \sqrt[3]{x^2 y}$

3. Take the Cube Root of the Perfect Cubes: Now, we take the cube root of the perfect cube terms. Remember that $\sqrt[3]{a^3} = a$, $\sqrt[3]{x^6} = x^2$, and $\sqrt[3]{y^{18}} = y^6$.

   $3x^2y^6 \sqrt[3]{x^2y}$

Therefore, the simplified radical form of $\sqrt[3]{27x^8 y^{19}}$ is $3x^2y^6 \sqrt[3]{x^2y}$.

Detailed Explanation

Let's deep dive into each step to ensure crystal clarity. Our mission is to simplify $\sqrt[3]{27x^8 y^{19}}$.

Factoring the Radicand: The Heart of Simplification

The radicand, $27x^8 y^{19}$, is where the magic begins. We need to decompose this into factors, specifically looking for perfect cubes because we're dealing with a cube root. Think of it like treasure hunting; perfect cubes are the valuable gems we want to extract.

• The Number 27: This is the easiest part. We recognize that $27 = 3 \times 3 \times 3 = 3^3$. Boom! A perfect cube right off the bat.

• The Variable $x^8$: Now, we need to express $x^8$ in terms of a multiple of 3. We can write $x^8$ as $x^{6+2}$, which is the same as $x^6 \cdot x^2$. Why did we do that? Because $x^6$ is a perfect cube! It's $(x^2)^3$. So, $x^8 = (x^2)^3 \cdot x^2$.

• The Variable $y^{19}$: Similarly, we tackle $y^{19}$. We express it as $y^{18+1}$, which is $y^{18} \cdot y$. Again, we're looking for a multiple of 3. And $y^{18}$ is indeed a perfect cube: $(y^6)^3$. Therefore, $y^{19} = (y^6)^3 \cdot y$.

Putting it all together, we rewrite the original radicand as:

$27x^8 y^{19} = 3^3 \cdot (x^2)^3 \cdot x^2 \cdot (y^6)^3 \cdot y$

Separating Perfect Cubes: Isolating the Gems

Now we strategically regroup terms to separate out the perfect cubes:

$\sqrt[3]{27x^8 y^{19}} = \sqrt[3]{3^3 \cdot (x^2)^3 \cdot (y^6)^3 \cdot x^2 \cdot y}$

Using the property $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, we can split the radical into two parts:

$\sqrt[3]{3^3 \cdot (x^2)^3 \cdot (y^6)^3} \cdot \sqrt[3]{x^2 \cdot y}$

The first radical contains all the perfect cubes, and the second radical contains the remaining terms that cannot be simplified further in terms of cube roots.

Taking the Cube Root: Extracting the Simplified Terms

This is where we finally extract the cube roots of the perfect cubes. Remember, $\sqrt[3]{a^3} = a$.

• $\sqrt[3]{3^3} = 3$
• $\sqrt[3]{(x^2)^3} = x^2$
• $\sqrt[3]{(y^6)^3} = y^6$

So, the first radical simplifies to $3x^2y^6$.

The second radical, $\sqrt[3]{x^2y}$, cannot be simplified further because $x^2$ and $y$ don't have factors that are perfect cubes.

The Final Result: The Simplified Radical Form

Combining the simplified terms, we get the final answer:

$3x^2y^6 \sqrt[3]{x^2y}$

This is the expression $\sqrt[3]{27x^8 y^{19}}$ written in its simplest radical form. We've successfully extracted all possible perfect cubes from the radicand, leaving us with a simplified expression.

Why This Matters

Simplifying radicals isn't just a mathematical exercise; it's a crucial skill in various fields, including physics, engineering, and computer science. Simplified expressions are easier to work with, prevent errors, and can reveal underlying patterns that might be obscured in a more complex form.

Moreover, understanding how to manipulate radicals and exponents strengthens your overall mathematical foundation, preparing you for more advanced concepts like calculus and differential equations.

So, while it might seem like a small detail, mastering radical simplification is a significant step towards becoming a proficient problem-solver in STEM fields.

Practice Problems

To solidify your understanding, try simplifying these expressions:

1. $\sqrt[3]{81a^{10}b^5}$
2. $\sqrt[3]{64x^7y^{12}}$
3. $\sqrt[3]{125p^4q^{16}}$

Work through these problems step-by-step, and compare your answers with solutions to ensure you've grasped the concepts. Good luck, and happy simplifying! Remember, practice makes perfect, so don't be afraid to tackle more challenging problems as you become more comfortable with the process.

By mastering the art of simplifying radicals, you'll not only enhance your mathematical skills but also gain a deeper appreciation for the elegance and power of mathematical notation. Keep practicing, keep exploring, and you'll find that math can be both challenging and rewarding.