https://file.aiquickdraw.com/mathbot/file/1725795504401k9xypk6l.jpg

The problem from the image states:

Given \( x = 4^7 \cdot 5^{14} \), what is the value of \( x^{\frac{1}{4}} \)?

### Step-by-step solution:

1. **Expression for \(x\)**: 
   \[
   x = 4^7 \cdot 5^{14}
   \]
   We can rewrite \( 4^7 \) as \( (2^2)^7 = 2^{14} \). Thus:
   \[
   x = 2^{14} \cdot 5^{14}
   \]
   This can be factored as:
   \[
   x = (2 \cdot 5)^{14} = 10^{14}
   \]

2. **Now, we need to calculate \( x^{\frac{1}{4}} \)**:
   \[
   x^{\frac{1}{4}} = (10^{14})^{\frac{1}{4}} = 10^{14 \cdot \frac{1}{4}} = 10^{3.5}
   \]

3. **Approximating \(10^{3.5}\)**:
   \[
   10^{3.5} = \sqrt{10^7}
   \]
   \[
   \sqrt{10^7} \approx 3.16 \times 10^3 = 3160
   \]
   Now, we need the approximate value of \(3160 \div 10 = 100\).

Thus, the correct answer is:
\[
\boxed{100}
\]

Would you like more details or have any questions?

### Related questions:
1. How would you solve \( x^{1/3} \) if \( x = 8^6 \)?
2. What happens if the power was \( \frac{1}{5} \) instead of \( \frac{1}{4} \)?
3. How would the calculation change if \( x = 6^7 \cdot 5^{14} \)?
4. How does the rule of exponents help simplify such expressions?
5. How does approximating square roots affect the accuracy of such solutions?

**Tip**: When dealing with powers of 10, it's often useful to simplify using scientific notation to make the calculation easier.

Jika x = 4^7 ⋅ 5^14, maka nilai dari x^(1/4) adalah?

This problem explores how to simplify and solve the expression x^(1/4) given that x = 4^7 ⋅ 5^14. It involves exponent rules and approximating square roots. Suitable for students in Grades 9-12.

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