https://r2.erweima.ai/mathbot/image/1724281812228l9m2e5od.jpg

The expression in the image is:

\[
\frac{\sqrt[5]{2^{10}}}{4} - \frac{3}{\sqrt[8]{3^{16}}}
\]

Let's simplify this expression step by step.

### Step 1: Simplify the Radicals

1. \(\sqrt[5]{2^{10}}\):
   \[
   \sqrt[5]{2^{10}} = 2^{\frac{10}{5}} = 2^2 = 4
   \]

2. \(\sqrt[8]{3^{16}}\):
   \[
   \sqrt[8]{3^{16}} = 3^{\frac{16}{8}} = 3^2 = 9
   \]

### Step 2: Substitute Back into the Expression

Now, substitute the simplified radicals back into the original expression:

\[
\frac{4}{4} - \frac{3}{9}
\]

### Step 3: Simplify the Fractions

1. \(\frac{4}{4} = 1\)
2. \(\frac{3}{9} = \frac{1}{3}\)

So the expression becomes:

\[
1 - \frac{1}{3}
\]

### Step 4: Perform the Subtraction

To subtract, convert \(1\) to a fraction with the same denominator as \(\frac{1}{3}\):

\[
1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\]

### Final Answer

The simplified value of the expression is:

\[
\boxed{\frac{2}{3}}
\]

Would you like more details or have any questions? Here are some related questions to explore:

1. What are the general rules for simplifying radicals with different indices?
2. How do you convert a whole number to a fraction to perform subtraction?
3. What are the properties of exponents that allow us to simplify \(a^{\frac{m}{n}}\)?
4. How can you simplify an expression involving radicals without changing the base numbers?
5. What methods can be used to simplify complex fractions involving roots?

**Tip:** When dealing with radicals, always check if the exponent can be simplified by dividing it by the root index before performing other operations.

Learn how to simplify the expression \(\frac{\sqrt[5]{2^{10}}}{4} - \frac{3}{\sqrt[8]{3^{16}}}\) step-by-step with detailed explanations and examples. Understand radicals, fractional exponents, and simplifying complex fractions involving roots. Suitable for Grades 9-12.

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