https://r2.erweima.ai/mathbot/image/1725680202514pud2wpok.pdf

Here are the algebraic expressions from the uploaded file, along with their simplifications:

1) **(3y²)(4y⁵)**  
   Simplification: \( 3 \times 4 = 12 \) and \( y^2 \times y^5 = y^{2+5} = y^7 \)  
   Result: \( 12y^7 \)

2) **\(\frac{a^9a^{-2}}{a}\)**  
   Simplification: \( a^9 \times a^{-2} = a^{9-2} = a^7 \), and dividing by \( a \) gives \( a^{7-1} = a^6 \)  
   Result: \( a^6 \)

3) **\(\frac{z^2z^4}{z^3z^{-1}}\)**  
   Simplification: \( z^2 \times z^4 = z^6 \) and \( z^3 \times z^{-1} = z^{3-1} = z^2 \), so \( \frac{z^6}{z^2} = z^{6-2} = z^4 \)  
   Result: \( z^4 \)

4) **\((2y²)³\)**  
   Simplification: Apply the power to both the coefficient and the variable: \( 2^3 = 8 \) and \( (y^2)^3 = y^{2 \times 3} = y^6 \)  
   Result: \( 8y^6 \)

5) **\(\left(\frac{a²}{4}\right)^3\)**  
   Simplification: \( a^2 \) raised to the third power gives \( a^{2 \times 3} = a^6 \), and \( 4^3 = 64 \)  
   Result: \( \frac{a^6}{64} \)

6) **\((\sqrt{\frac{b^3}{4}})(\sqrt{b})\)**  
   Simplification: \( \sqrt{\frac{b^3}{4}} = \frac{\sqrt{b^3}}{2} = \frac{b^{3/2}}{2} \), and \( \sqrt{b} = b^{1/2} \), so multiplying gives \( \frac{b^{3/2} \times b^{1/2}}{2} = \frac{b^{(3/2 + 1/2)}}{2} = \frac{b^2}{2} \)  
   Result: \( \frac{b^2}{2} \)

7) **\((2\sqrt{a})(\sqrt{a²/3})\)**  
   Simplification: \( 2\sqrt{a} = 2a^{1/2} \) and \( \sqrt{a^2/3} = \frac{a}{\sqrt{3}} \), so multiplying gives \( \frac{2a^{1/2} \times a}{\sqrt{3}} = \frac{2a^{(1/2 + 1)}}{\sqrt{3}} = \frac{2a^{3/2}}{\sqrt{3}} \)  
   Result: \( \frac{2a^{3/2}}{\sqrt{3}} \)

8) **\(\sqrt{y}\sqrt{y^3}\)**  
   Simplification: \( \sqrt{y} = y^{1/2} \) and \( \sqrt{y^3} = y^{3/2} \), so multiplying gives \( y^{1/2 + 3/2} = y^2 \)  
   Result: \( y^2 \)

9) **\(\sqrt{\frac{16a^3b}{ab^5}}\)**  
   Simplification: \( \frac{16a^3b}{ab^5} = 16a^{3-1}b^{1-5} = 16a^2b^{-4} \), so \( \sqrt{16a^2b^{-4}} = 4a\sqrt{b^{-4}} = 4a\frac{1}{b^2} \)  
   Result: \( \frac{4a}{b^2} \)

10) **\(\sqrt{\frac{54x^2y^4}{2x^5y^3}}\)**  
    Simplification: \( \frac{54x^2y^4}{2x^5y^3} = \frac{54}{2} \times \frac{x^2}{x^5} \times \frac{y^4}{y^3} = 27 \times x^{2-5} \times y^{4-3} = 27x^{-3}y \), so \( \sqrt{27x^{-3}y} = \sqrt{27} \times \sqrt{x^{-3}} \times \sqrt{y} = 3\sqrt{3} \times \frac{1}{x^{3/2}} \times y^{1/2} \)  
    Result: \( \frac{3\sqrt{3}y^{1/2}}{x^{3/2}} \)

If you need further clarification or details, feel free to ask! Here are 5 related questions to expand your understanding:

1. How do you simplify expressions with fractional exponents?
2. How do you multiply and divide powers of the same base?
3. What are the rules for simplifying radical expressions?
4. How do you apply exponents to both the numerator and denominator in a fraction?
5. How do you handle negative exponents during simplification?

**Tip:** Always combine terms with the same base before simplifying further, especially when dealing with radicals or fractional exponents.

Simplify the following algebraic expressions, eliminating any negative exponents.

Learn how to simplify algebraic expressions involving exponents, radicals, and negative powers. This guide provides step-by-step solutions for expressions like (3y²)(4y⁵) and √(16a³b / ab⁵), suitable for high school students in Grades 9-11.

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