Math Problem: Simplify Expression To $9 \sqrt[3]{10}$

Hey math whizzes and problem solvers! Today, we're diving into a cool question that's all about simplifying expressions. We've got a target value, $9$, and we need to figure out which of the given options actually equals this. This is a fantastic way to test your understanding of radicals and how to combine them. So, grab your calculators (or just your sharp minds!), and let's break down why one of these expressions is the true match for $9$.

Understanding Radicals and Combining Like Terms

Before we jump into the options, let's quickly recap what we're dealing with. We're seeing a cube root, represented by the $\sqrt[3]{}$ symbol. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. In our problem, we're working with the cube root of 10, which is $\sqrt[3]{10}$. This number can't be simplified further into a whole number, so we'll be keeping it as is.

Now, the key to solving this problem lies in the concept of combining like terms. Think of it like adding apples and apples, or oranges and oranges. You can't just add 5 apples and 3 oranges and say you have 8 apple-oranges; you still have 5 apples and 3 oranges. The same logic applies to radicals. You can only add or subtract terms that have the same radical part. For instance, you can add $2 \sqrt{5}$ and $3 \sqrt{5}$ to get $5 \sqrt{5}$, because both terms have $\sqrt{5}$. However, you cannot add $2 \sqrt{5}$ and $3 \sqrt{7}$ directly because the radical parts ($\sqrt{5}$ and $\sqrt{7}$) are different.

In our specific problem, the target expression is $9$. This means we're looking for an expression that, when simplified, results in a coefficient of 9 multiplied by the cube root of 10. The radicals in the options are a mix of square roots ($\sqrt{}$) and cube roots ($\sqrt[3]{}$). This distinction is crucial! A square root of 10 is $\sqrt{10}$, and a cube root of 10 is $\sqrt[3]{10}$. These are fundamentally different numbers. Therefore, to combine terms, we need both the coefficient and the radical part to be identical.

Let's consider the target expression $9$. The number 9 is the coefficient, and $\sqrt[3]{10}$ is the radical part. We need to find an option where adding the terms results in exactly this combination.

Analyzing the Options: A Step-by-Step Approach

Alright guys, let's put on our detective hats and examine each option to see which one holds the key to $9$.

Option A: $5 \sqrt{10} + 4 \sqrt{10}$

In this option, we have two terms, both involving the square root of 10 ($\sqrt{10}$). Since the radical part is the same ($\sqrt{10}$) in both terms, we can combine them. We simply add the coefficients: $5 + 4 = 9$. So, this expression simplifies to $9 \sqrt{10}$. Now, compare this to our target, $9$. The radical part here is $\sqrt{10}$ (square root), while our target has $\sqrt[3]{10}$ (cube root). Since $\sqrt{10} \neq \sqrt[3]{10}$, Option A is not the correct answer. It's close because the coefficient is right, but the radical type is wrong.

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

Let's look at this one. We have two terms, and both have the cube root of 10 ($\sqrt[3]{10}$) as their radical part. This is exactly what we're looking for! Since the radical parts are identical, we can combine the terms by adding their coefficients: $5 + 4 = 9$. This gives us $9 \sqrt[3]{10}$. Bingo! This expression perfectly matches our target value. Therefore, Option B is the correct answer. High five!

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

Here, we have a mix of radicals. The first term has $5 \sqrt{10}$ (square root of 10), and the second term has $4 \sqrt[3]{10}$ (cube root of 10). As we discussed earlier, you can only combine terms with identical radical parts. Since $\sqrt{10}$ and $\sqrt[3]{10}$ are different, these terms are unlike terms. We cannot add the coefficients (5 and 4) because the radical parts don't match. This expression cannot be simplified further into a single term like $9$. Thus, Option C is incorrect.

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

Similar to Option C, this option also presents a mix of different types of radicals. We have $5 \sqrt[3]{10}$ (cube root of 10) and $4 \sqrt{10}$ (square root of 10). Because the radical parts, $\sqrt[3]{10}$ and $\sqrt{10}$, are not the same, these are unlike terms. We cannot combine them by adding their coefficients. This expression also cannot be simplified to the form $9$. So, Option D is incorrect.

The Winning Expression

After carefully examining all the options, we found that Option B: $5 \sqrt[3]{10} + 4 \sqrt[3]{10}$ is the only expression that simplifies to $9$. This is because both terms contain the same radical part, $\sqrt[3]{10}$, allowing us to add their coefficients, 5 and 4, to get 9. The other options failed because they either had the wrong type of radical (square root instead of cube root) or a mix of unlike radicals that couldn't be combined.

Keep practicing these types of problems, guys! Understanding the rules of combining like terms with radicals is super important for more advanced math. You've got this!