https://r2.erweima.ai/mathbot/image/1726610768262mw0hbgp0.jpg

Let's work through simplifying the expressions from the image, step by step, while eliminating any negative exponents. I'll start with the first few problems and walk you through the solutions.

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**Problem 61 (a)**:
\[
x^{3/4} \cdot x^{-5/4}
\]
To simplify, we add the exponents since the bases are the same:
\[
x^{3/4 + (-5/4)} = x^{-2/4} = x^{-1/2}
\]
Now, eliminate the negative exponent:
\[
\frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}}
\]
**Final answer**: \(\frac{1}{\sqrt{x}}\)

---

**Problem 61 (b)**:
\[
y^{2/3} \cdot y^{-4/3}
\]
Again, add the exponents:
\[
y^{2/3 + (-4/3)} = y^{-2/3}
\]
Eliminate the negative exponent:
\[
\frac{1}{y^{2/3}} = \frac{1}{\sqrt[3]{y^2}}
\]
**Final answer**: \(\frac{1}{\sqrt[3]{y^2}}\)

---

**Problem 62 (a)**:
\[
(4b)^{1/2} \cdot (8b^{1/4})
\]
First, simplify each term:
\[
(4b)^{1/2} = 4^{1/2} \cdot b^{1/2} = 2 \cdot b^{1/2}
\]
Now, multiply by the second part:
\[
2b^{1/2} \cdot 8b^{1/4} = (2 \cdot 8) \cdot (b^{1/2} \cdot b^{1/4}) = 16 \cdot b^{1/2 + 1/4} = 16b^{3/4}
\]
**Final answer**: \(16b^{3/4}\)

---

**Problem 62 (b)**:
\[
(3a^{3/4})^2 \cdot (5a^{1/2})
\]
First, square the first term:
\[
(3a^{3/4})^2 = 9a^{3/2}
\]
Now, multiply by the second part:
\[
9a^{3/2} \cdot 5a^{1/2} = (9 \cdot 5) \cdot a^{3/2 + 1/2} = 45a^2
\]
**Final answer**: \(45a^2\)

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Let me know if you'd like further details or clarifications on these steps! Now, here are some questions to deepen your understanding:

1. How do you add exponents when multiplying two terms with the same base?
2. What happens when you have a negative exponent, and how do you eliminate it?
3. How do you apply exponents when distributing across multiplication (like \((ab)^n\))?
4. What steps do you take when simplifying an expression with fractional exponents?
5. How can you rewrite an expression involving radicals in terms of exponents?

**Tip**: When dealing with fractional exponents, remember that the numerator represents the power, and the denominator represents the root!

Simplifique la expresión y elimine cualquier exponente negativo(s). Suponga que todas las letras denotan números positivos.

This guide helps students simplify expressions with negative and fractional exponents, providing a step-by-step breakdown. Learn key exponent laws and see how to eliminate negative exponents in algebraic expressions.

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