Solving $9 \sqrt[3]{10}$ Expression A Step-by-Step Guide

Hey guys! Let's dive into the world of mathematical expressions and tackle a problem that might seem tricky at first glance. We're going to break down the expression $9 \sqrt[3]{10}$ and figure out which of the given options is equivalent. This isn't just about finding the right answer; it's about understanding the underlying principles of radicals and how they work. So, buckle up and let's get started!

Understanding the Basics of Radicals

Before we jump into the problem, let's quickly review what radicals are and how they behave. Radicals, in their simplest form, are ways of representing roots of numbers. The most common radical is the square root ($\sqrt{}$), but we also have cube roots ($\sqrt[3]{}$), fourth roots ($\sqrt[4]{}$), and so on. The little number nestled in the crook of the radical symbol is called the index, and it tells us what root we're dealing with. For example, $\sqrt[3]{8}$ means "the cube root of 8," which is 2 because $2 \* 2 \* 2 = 8$.

The expression we're working with, $9 \sqrt[3]{10}$, involves a cube root. This means we're looking for a number that, when multiplied by itself three times, equals 10. Since 10 isn't a perfect cube (like 8 or 27), its cube root is an irrational number, meaning it can't be expressed as a simple fraction. This is why we leave it in radical form as $\sqrt[3]{10}$. The 9 in front of the radical is a coefficient, simply indicating that we have 9 of these $\sqrt[3]{10}$ terms.

Key Concepts to Remember:

• Radical: A symbol ($\sqrt{}$) that indicates a root of a number.
• Index: The small number in the radical symbol indicating the type of root (e.g., 3 for cube root).
• Coefficient: The number in front of the radical, indicating how many of that radical term we have.
• Like Radicals: Radicals with the same index and radicand (the number under the radical symbol).

Addition and Subtraction of Radicals

Now, let's talk about how we can add or subtract radicals. Just like we can only combine like terms in algebra (e.g., $3x + 2x = 5x$), we can only add or subtract radicals if they are like radicals. This means they must have the same index and the same radicand. For example, $2\sqrt{5}$ and $7\sqrt{5}$ are like radicals because they both have an index of 2 (square root) and a radicand of 5. We can add them together: $2\sqrt{5} + 7\sqrt{5} = 9\sqrt{5}$.

However, $2\sqrt{5}$ and $7\sqrt{3}$ are not like radicals because they have different radicands. Similarly, $2\sqrt{5}$ and $7\sqrt[3]{5}$ are not like radicals because they have different indices. We cannot directly add or subtract these radicals.

In our problem, we're dealing with the expression $9 \sqrt[3]{10}$. This means we're looking for an option that, when simplified, results in 9 cube roots of 10. Keep this in mind as we analyze the answer choices.

Analyzing the Answer Choices

Okay, let's get down to business and examine the answer choices provided. We'll go through each one, step by step, and see if it simplifies to $9 \sqrt[3]{10}$. Remember, the key is to identify like radicals and combine them correctly.

A. $5 \sqrt{10} + 4 \sqrt{10}$

In this option, we have two terms, both involving square roots of 10. Notice that the index is 2 (square root), and the radicand is 10 in both terms. This means they are like radicals, and we can combine them. Adding the coefficients, we get:

$5 \sqrt{10} + 4 \sqrt{10} = (5 + 4) \sqrt{10} = 9 \sqrt{10}$

This result is $9 \sqrt{10}$, which involves a square root, not a cube root. So, option A is not equivalent to our target expression, $9 \sqrt[3]{10}$. We can eliminate it.

B. $5 \sqrt[3]{10} + 4 \sqrt[3]{10}$

Here, we have two terms involving cube roots of 10. Both terms have an index of 3 (cube root) and a radicand of 10. These are like radicals! Let's add them up:

$5 \sqrt[3]{10} + 4 \sqrt[3]{10} = (5 + 4) \sqrt[3]{10} = 9 \sqrt[3]{10}$

Bingo! This simplifies exactly to $9 \sqrt[3]{10}$, which is the expression we're looking for. So, option B is the correct answer. But, just to be thorough, let's take a look at the remaining options.

C. $5 \sqrt{10} + 4 \sqrt[3]{10}$

In this case, we have a square root of 10 and a cube root of 10. The indices are different (2 and 3), so these are not like radicals. We cannot combine these terms any further. The expression remains $5 \sqrt{10} + 4 \sqrt[3]{10}$, which is clearly not equal to $9 \sqrt[3]{10}$. Option C is incorrect.

D. $5 \sqrt[3]{10} + 4 \sqrt{10}$

Similar to option C, this option has a mix of cube roots and square roots. We have $5 \sqrt[3]{10}$ and $4 \sqrt{10}$. Again, the indices are different, so these are not like radicals, and we cannot combine them. The expression stays as $5 \sqrt[3]{10} + 4 \sqrt{10}$, which is not the same as $9 \sqrt[3]{10}$. Option D is also incorrect.

Why is Option B Correct?

Option B, $5 \sqrt[3]{10} + 4 \sqrt[3]{10}$, is the only expression that simplifies to $9 \sqrt[3]{10}$. This is because it's the only option with like radicals that, when combined, give us the desired result. We have 5 cube roots of 10, and we're adding 4 more cube roots of 10, giving us a total of 9 cube roots of 10.

Step-by-Step Solution

Let's recap the solution step-by-step to make sure we've got it all down:

1. Understand the problem: We need to find the expression that is equivalent to $9 \sqrt[3]{10}$.
2. Review radical basics: Remember that we can only add or subtract like radicals (same index and radicand).
3. Analyze option A: $5 \sqrt{10} + 4 \sqrt{10} = 9 \sqrt{10}$. This is not $9 \sqrt[3]{10}$, so it's incorrect.
4. Analyze option B: $5 \sqrt[3]{10} + 4 \sqrt[3]{10} = 9 \sqrt[3]{10}$. This matches our target expression, so it's the correct answer.
5. Analyze option C: $5 \sqrt{10} + 4 \sqrt[3]{10}$. These are not like radicals, so we can't combine them. Incorrect.
6. Analyze option D: $5 \sqrt[3]{10} + 4 \sqrt{10}$. These are also not like radicals. Incorrect.
7. Conclude: Option B is the correct answer.

Key Takeaways

So, what have we learned from this exercise? Here are some key takeaways:

• Like radicals are essential for addition and subtraction. You can only combine radicals that have the same index and radicand.
• Pay attention to the index. Is it a square root, a cube root, or something else? This makes a big difference.
• Simplify expressions carefully. Take it one step at a time, and don't try to rush the process.
• Double-check your work. It's always a good idea to go back and make sure your answer makes sense in the context of the problem.

Practice Makes Perfect

Guys, the best way to master radical expressions is to practice! Try working through similar problems, and don't be afraid to make mistakes. Every mistake is a learning opportunity. You can find plenty of practice problems online or in textbooks. The more you work with radicals, the more comfortable you'll become with them.

Conclusion: Option B is the Winner!

In conclusion, the expression that equals $9 \sqrt[3]{10}$ is B. $5 \sqrt[3]{10} + 4 \sqrt[3]{10}$. We arrived at this answer by understanding the properties of radicals, identifying like radicals, and carefully simplifying the expressions. Keep practicing, and you'll become a radical master in no time!