Simplifying Expressions With Radicals And Exponents

Hey guys! Today, we're diving into the exciting world of simplifying expressions that involve both radicals and exponents. Specifically, we're going to tackle a problem where we need to simplify a complex expression involving fractional exponents and radicals. This is a common type of problem in algebra, and mastering it will definitely boost your math skills. So, let's jump right in and break it down step by step.

Understanding the Problem

Before we dive into the solution, let's make sure we fully grasp the problem. We're given an expression with radicals and fractional exponents:

$\sqrt[3]{(\frac{a^{\frac{3}{2}}}{b^{\frac{1}{3}}})^2} : \sqrt[3]{(\frac{1}{a^2})^{-2} b^{\frac{4}{3}}}$

where a and b are positive real numbers. Our mission, should we choose to accept it (and we do!), is to simplify this expression into its most basic form. This involves using the properties of exponents and radicals to manipulate the expression until we can't simplify it any further. We'll need to remember things like how to deal with fractional exponents, negative exponents, and how radicals relate to exponents. It might seem daunting at first, but don't worry, we'll take it one step at a time and make sure everything is crystal clear.

Remember, the key to simplifying complex expressions is to break them down into smaller, more manageable parts. We'll focus on simplifying the terms inside the radicals first, then deal with the radicals themselves. We'll also pay close attention to the order of operations, ensuring we're performing the correct operations in the right sequence. So, with our thinking caps on, let's begin the simplification journey!

Breaking Down the Expression

Okay, let's start by simplifying the expression step-by-step. The key here is to take it slow and apply the rules of exponents and radicals carefully. Remember, patience is a virtue, especially in math!

Step 1: Simplifying Inside the First Radical

Let's focus on the first part of the expression: $\sqrt[3]{(\frac{a^{\frac{3}{2}}}{b^{\frac{1}{3}}})^2}$. We need to simplify the fraction inside the radical first, and then deal with the outer exponent of 2. When we have a power raised to another power, we multiply the exponents. So,

$(\frac{a^{\frac{3}{2}}}{b^{\frac{1}{3}}})^2 = \frac{(a^{\frac{3}{2}})^2}{(b^{\frac{1}{3}})^2}$

Now, we multiply the exponents:

$\frac{a^{\frac{3}{2} \cdot 2}}{b^{\frac{1}{3} \cdot 2}} = \frac{a^3}{b^{\frac{2}{3}}}$

So, the first radical now looks like this: $\sqrt[3]{\frac{a^3}{b^{\frac{2}{3}}}}$. We're making progress, guys! Notice how we're breaking down the problem into smaller, simpler pieces. This is a powerful technique for tackling any complex math problem.

Step 2: Simplifying Inside the Second Radical

Now, let's tackle the second radical: $\sqrt[3]{(\frac{1}{a^2})^{-2} b^{\frac{4}{3}}}$. Again, we'll simplify the inside first. We have a term with a negative exponent, $(\frac{1}{a^2})^{-2}$. Remember, a negative exponent means we take the reciprocal of the base and make the exponent positive. So,

$(\frac{1}{a^2})^{-2} = (a^{-2})^{-2} = a^{(-2) \cdot (-2)} = a^4$

Now we can substitute this back into the second radical:

$\sqrt[3]{a^4 b^{\frac{4}{3}}}$

Awesome! We've simplified the expressions inside both radicals. We're one step closer to the final answer. Feels good, doesn't it?

Combining and Simplifying Radicals

Now that we've simplified the expressions inside the radicals, let's combine them and see what we get. Remember, our original problem was:

$\sqrt[3]{(\frac{a^{\frac{3}{2}}}{b^{\frac{1}{3}}})^2} : \sqrt[3]{(\frac{1}{a^2})^{-2} b^{\frac{4}{3}}}$

We've simplified the insides, so now it looks like this:

$\sqrt[3]{\frac{a^3}{b^{\frac{2}{3}}}} : \sqrt[3]{a^4 b^{\frac{4}{3}}}$

The colon (:) represents division, so we can rewrite this as a fraction:

$\frac{\sqrt[3]{\frac{a^3}{b^{\frac{2}{3}}}}}{\sqrt[3]{a^4 b^{\frac{4}{3}}}}$

When we divide radicals with the same index (in this case, a cube root), we can combine them under a single radical:

\sqrt[3]{\frac{\frac{a^3}{b^{\frac{2}{3}}}}{a^4 b^{\frac{4}{3}}}}=\sqrt[3]{\frac{a^3}{b^{\frac{2}{3}}} imes rac{1}{a^4 b^{\frac{4}{3}}}}

Step 3: Simplifying the Fraction Under the Radical

Now, we need to simplify the fraction inside the cube root. This involves using the rules of exponents for division. When we divide terms with the same base, we subtract the exponents:

$\frac{a^3}{a^4} = a^{3-4} = a^{-1}$

And for the b terms:

\frac{1}{b^{\frac{2}{3}} b^{\frac{4}{3}}} = rac{1}{b^{\frac{2}{3} + \frac{4}{3}}} = rac{1}{b^{\frac{6}{3}}} = rac{1}{b^2}

Putting it all together, the fraction inside the cube root becomes:

$a^{-1} \cdot \frac{1}{b^2} = \frac{a^{-1}}{b^2} = \frac{1}{a b^2}$

So our expression now looks like this:

$\sqrt[3]{\frac{1}{a b^2}}$

We're almost there! Just a little more simplification to go.

Final Simplification and the Answer

We're at the final stage, guys! We have:

$\sqrt[3]{\frac{1}{a b^2}}$

To simplify this further, we can rewrite the cube root as a fractional exponent:

$(\frac{1}{a b^2})^{\frac{1}{3}}$

Now, we can distribute the exponent to each term in the fraction:

$\frac{1^{\frac{1}{3}}}{(a b^2)^{\frac{1}{3}}} = \frac{1}{a^{\frac{1}{3}} (b^2)^{\frac{1}{3}}} = \frac{1}{a^{\frac{1}{3}} b^{\frac{2}{3}}}$

Rationalizing the Denominator (Optional, but Good Practice)

Sometimes, it's helpful to rationalize the denominator, which means getting rid of any radicals in the denominator. To do this, we want to multiply the numerator and denominator by a term that will make the exponents in the denominator whole numbers. In this case, we need to multiply by $a^{\frac{2}{3}}$ to make the exponent of a equal to 1, and by $b^{\frac{1}{3}}$ to make the exponent of b equal to 1. So, we multiply both the numerator and the denominator by $a^{\frac{2}{3}}b^{\frac{1}{3}}$:

$\frac{1}{a^{\frac{1}{3}} b^{\frac{2}{3}}} \cdot \frac{a^{\frac{2}{3}}b^{\frac{1}{3}}}{a^{\frac{2}{3}}b^{\frac{1}{3}}} = \frac{a^{\frac{2}{3}}b^{\frac{1}{3}}}{a^{\frac{1}{3} + \frac{2}{3}} b^{\frac{2}{3} + \frac{1}{3}}} = \frac{a^{\frac{2}{3}}b^{\frac{1}{3}}}{a^1 b^1} = \frac{a^{\frac{2}{3}}b^{\frac{1}{3}}}{ab}$

This can also be written as:

$\frac{\sqrt[3]{a^2b}}{ab}$

The Final Simplified Form

However, going back to our previous simplified form, $\frac{1}{a^{\frac{1}{3}} b^{\frac{2}{3}}}$, we can also express it using radicals:

$\frac{1}{\sqrt[3]{a b^2}}$

Depending on the format you need the answer in, either $\frac{1}{\sqrt[3]{a b^2}}$ or $\frac{\sqrt[3]{a^2b}}{ab}$ could be the final simplified form.

Conclusion and Key Takeaways

Wow, guys, we did it! We successfully simplified a complex expression with radicals and fractional exponents. Give yourselves a pat on the back! The key to solving these kinds of problems is to:

1. Break it down: Divide the problem into smaller, more manageable steps.
2. Know your rules: Remember the properties of exponents and radicals.
3. Be patient: Take your time and work through each step carefully.

We started with a seemingly complicated expression and, by applying the rules of exponents and radicals systematically, we arrived at a simplified form. This process highlights the power of breaking down complex problems into smaller, more digestible steps.

Remember, practice makes perfect. The more you work with these types of expressions, the more comfortable you'll become with simplifying them. So, keep practicing, and you'll be a master of radicals and exponents in no time! If you have any questions or want to try another example, just let me know. Keep up the awesome work! ✨