https://r2.erweima.ai/mathbot/image/1726752425508co90ab75.jpg

The expression provided in the image is:

\[
\sqrt[4]{\frac{27 \times 10^5}{0.3 \times 10^2}}
\]

Let's simplify this step by step.

### Step 1: Simplify the fraction inside the radical
\[
\frac{27 \times 10^5}{0.3 \times 10^2} = \frac{27}{0.3} \times \frac{10^5}{10^2}
\]

First, simplify the numerical fraction:
\[
\frac{27}{0.3} = 90
\]

Now simplify the powers of 10:
\[
\frac{10^5}{10^2} = 10^{5-2} = 10^3
\]

Thus, the expression becomes:
\[
90 \times 10^3 = 9 \times 10^4
\]

### Step 2: Apply the fourth root
Now take the fourth root of \(9 \times 10^4\):
\[
\sqrt[4]{9 \times 10^4}
\]

This can be split as:
\[
\sqrt[4]{9} \times \sqrt[4]{10^4}
\]

We know:
\[
\sqrt[4]{9} = 9^{1/4} \quad \text{(which is approximately 1.732)}
\]
and
\[
\sqrt[4]{10^4} = 10^{4/4} = 10
\]

So the final result is:
\[
1.732 \times 10 = 17.32
\]

Thus, the value of the expression is approximately **17.32**.

Would you like more details or clarifications on any step?

Here are some related questions to explore further:
1. How do you handle different powers of 10 in similar expressions?
2. How can you simplify expressions involving roots and exponents?
3. What are some common mistakes to avoid when working with powers of 10?
4. Can you generalize this process to other roots, such as cube roots?
5. What other methods can be used to approximate roots, like using logarithms?

**Tip**: Always simplify the numerical and exponential parts separately when dealing with complex expressions!

Simplify the expression: \[\sqrt[4]{\frac{27 \times 10^5}{0.3 \times 10^2}}\]

Learn how to simplify the expression \(\sqrt[4]{\frac{27 \times 10^5}{0.3 \times 10^2}}\). This problem involves key concepts such as exponents, radical simplification, and powers of ten. Suitable for students in Grades 10-12.

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