Radical Notation: Simplifying (100/81)^(3/2)

Hey guys! Let's dive into how to express the expression $\left(\frac{100}{81}\right)^{3 / 2}$ using radical notation and simplify it. This is a fun problem that combines exponents and radicals, so buckle up!

Understanding the Basics

Before we jump into the specific problem, let's quickly review what radical notation is and how it relates to fractional exponents. You know, just to make sure we're all on the same page. Remember, mathematics is all about building on fundamental concepts.

What is Radical Notation?

Radical notation is a way of expressing roots using the radical symbol $\sqrt{}$. For example, $\sqrt{9}$ represents the square root of 9, which is 3. Similarly, $\sqrt[3]{8}$ represents the cube root of 8, which is 2. The small number above the radical symbol (like the 3 in $\sqrt[3]{8}$) is called the index, and it tells you what root you're taking. When there's no index written, it's assumed to be 2, meaning you're taking the square root.

Fractional Exponents and Radicals

Fractional exponents are closely related to radicals. A fractional exponent like $x^{a/b}$ can be rewritten in radical form as $\sqrt[b]{x^a}$ or $(\sqrt[b]{x})^a$. In other words, the denominator of the fraction becomes the index of the radical, and the numerator becomes the exponent of the radicand (the number inside the radical). This is super important and the key to solving our problem.

For example, $9^{1/2}$ is the same as $\sqrt{9}$, which equals 3. And $8^{2/3}$ can be written as $\sqrt[3]{8^2}$ or $(\sqrt[3]{8})^2$, both of which simplify to 4.

Converting (100/81)^(3/2) to Radical Notation

Now that we've got the basics down, let's tackle our problem: $\left(\frac{100}{81}\right)^{3 / 2}$.

Step 1: Identify the Components

In the expression $\left(\frac{100}{81}\right)^{3 / 2}$, we have a base of $\frac{100}{81}$ and an exponent of $\frac{3}{2}$. According to what we discussed, the denominator of the exponent (2) will be the index of our radical, and the numerator (3) will be the power to which we raise the radicand. Understanding this relationship is crucial for converting between fractional exponents and radicals.

Step 2: Rewrite in Radical Form

Using the relationship between fractional exponents and radicals, we can rewrite $\left(\frac{100}{81}\right)^{3 / 2}$ in radical notation as:

$\sqrt{\left(\frac{100}{81}\right)^3}$

Alternatively, we can write it as:

$\left(\sqrt{\frac{100}{81}}\right)^3$

Both forms are equivalent, but the second form is often easier to simplify because it involves taking the square root before cubing, which usually deals with smaller numbers. Choosing the right approach can save you time and effort.

Simplifying the Expression

Now that we've expressed the given expression in radical notation, let's simplify it. We'll use the second form, $\left(\sqrt{\frac{100}{81}}\right)^3$, as it's generally easier to work with.

Step 1: Simplify the Square Root

We need to find the square root of $\frac{100}{81}$. Remember that the square root of a fraction is the square root of the numerator divided by the square root of the denominator. So, we have:

$\sqrt{\frac{100}{81}} = \frac{\sqrt{100}}{\sqrt{81}}$

Since $\sqrt{100} = 10$ and $\sqrt{81} = 9$, we get:

$\frac{\sqrt{100}}{\sqrt{81}} = \frac{10}{9}$

Step 2: Cube the Result

Now that we've simplified the square root to $\frac{10}{9}$, we need to cube it. This means raising the fraction to the power of 3:

$\left(\frac{10}{9}\right)^3 = \frac{10^3}{9^3}$

Calculating the cubes, we have:

$\frac{10^3}{9^3} = \frac{1000}{729}$

Step 3: Final Simplified Form

So, the simplified form of $\left(\frac{100}{81}\right)^{3 / 2}$ is $\frac{1000}{729}$. And that's our final answer! We've successfully converted the expression to radical notation and simplified it.

Alternative Approach: Simplifying Before Converting

Just to show you there's often more than one way to skin a cat (or solve a math problem!), let's consider another approach. We could first cube the fraction and then take the square root.

Step 1: Cube the Fraction

$\left(\frac{100}{81}\right)^3 = \frac{100^3}{81^3} = \frac{1000000}{531441}$

Step 2: Take the Square Root

Now we need to find the square root of $\frac{1000000}{531441}$:

$\sqrt{\frac{1000000}{531441}} = \frac{\sqrt{1000000}}{\sqrt{531441}} = \frac{1000}{729}$

As you can see, we arrive at the same answer, $\frac{1000}{729}$. However, in this case, cubing the numbers first resulted in much larger numbers, which could be more challenging to work with if you don't have a calculator handy. This illustrates the importance of choosing the most efficient method.

Key Takeaways

1. Radical Notation: Understand how to convert fractional exponents to radical notation and vice versa. The denominator of the fractional exponent becomes the index of the radical.
2. Simplification: Always look for opportunities to simplify the expression before or after converting to radical notation. Simplifying early often makes the calculations easier.
3. Multiple Approaches: There are often multiple ways to solve a math problem. Experiment with different methods to find the one that works best for you.
4. Fractional Exponents: Grasp the concept that $x^{a/b}$ is equivalent to $\sqrt[b]{x^a}$ or $(\sqrt[b]{x})^a$. This is a cornerstone of working with radicals and exponents.

Practice Makes Perfect

The best way to master these concepts is to practice! Try converting other expressions with fractional exponents to radical notation and simplifying them. Here are a few examples to get you started:

• $\left(\frac{16}{25}\right)^{3 / 2}$
• $\left(\frac{8}{27}\right)^{2 / 3}$
• $49^{3 / 2}$

Work through these examples, and you'll become much more comfortable with radical notation and fractional exponents. Keep practicing, and you'll become a math whiz in no time! Understanding these concepts is super useful in algebra, calculus, and beyond, so it's totally worth the effort.

Conclusion

So there you have it! We've successfully expressed $\left(\frac{100}{81}\right)^{3 / 2}$ in radical notation and simplified it to $\frac{1000}{729}$. Remember to understand the relationship between fractional exponents and radicals, and always look for opportunities to simplify. Keep practicing, and you'll ace these problems every time. Happy calculating, folks! You've got this! And remember, math can actually be kinda fun when you get the hang of it! Cheers!