https://file.aiquickdraw.com/mathbot/file/1726911841977thiaxw5j.jpg

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### Left Side: **Rewrite these expressions using powers**
1. \(\sqrt{a}\) = \( a^{1/2} \)
2. \(^5\sqrt{c}\) = \( c^{1/5} \)
3. \(^7\sqrt{x^6}\) = \( x^{6/7} \)
4. \(\frac{1}{\sqrt{e}}\) = \( e^{-1/2} \)
5. \(\frac{1}{^5\sqrt{x^4}}\) = \( x^{-4/5} \)

### Right Side: **Simplify these expressions**
1. \(^6\sqrt{(8a^3)^6}\) = \( 8^1 a^3 = 8a^3 \)
2. \(^4\sqrt{\left(\frac{b^5}{2}\right)^9}\) = \( \left(\frac{b^5}{2}\right)^{9/4} = \frac{b^{45}}{2^{9/4}} \)
3. \(^4\sqrt{\frac{c^{23}}{c^7}}\) = \( \left(c^{23-7}\right)^{1/4} = c^{4} \)
4. \(^3\sqrt{8d^4 \times 27d^{-5}}\) = \( \sqrt[3]{216d^{-1}} = 6d^{-1/3} \)

Let me know if you need more details or clarifications!

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Here are 5 related questions to expand on the topic:
1. How do you simplify expressions involving fractional exponents?
2. What is the general rule for converting roots to exponents?
3. How do you handle negative exponents during simplifications?
4. Why does multiplying two powers of the same base result in adding exponents?
5. How do you simplify expressions involving cube roots?

**Tip:** Always express roots as fractional powers to make simplifying more intuitive!

Rewrite these expressions using powers and simplify these expressions as shown in the image.

Learn how to rewrite expressions using powers and simplify complex expressions involving roots and exponents. This problem demonstrates key algebraic concepts such as exponent laws and root-to-power conversion, suitable for students in Grades 9-11.

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