Simplify the given exponential expressions using the properties of exponents.

Let’s solve each problem one by one, using the properties of exponents.

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Problem 1: (a³)³ · 2a⁻¹

Step 1: Simplify (a³)³ → when you raise a power to a power, multiply the exponents:
→ a^(3×3) = a⁹

Step 2: Now we have: a⁹ · 2a⁻¹
Multiply coefficients: 1 × 2 = 2
Add exponents for same base ‘a’: a⁹ · a⁻¹ = a^(9 + (-1)) = a⁸

Final: 2a⁸

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Problem 2: (a⁴)⁻³ · 2a⁴

Step 1: (a⁴)⁻³ → multiply exponents: a^(4×-3) = a⁻¹²

Step 2: Now: a⁻¹² · 2a⁴
Coefficients: 1 × 2 = 2
Exponents: a⁻¹² · a⁴ = a^(-12+4) = a⁻⁸

Final: 2a⁻⁸ or 2/a⁸ (but since worksheet says “simplify”, and doesn’t specify positive exponents only, either is fine — but usually they want positive exponents if possible. Let’s check other problems — Problem 5 has negative exponents in answer? Actually, let’s keep as simplified expression. Since no instruction, we’ll leave as 2a⁻⁸ unless told otherwise. But looking at Problem 6, it has z⁰ which becomes 1, so maybe they accept negative exponents. We’ll go with 2a⁻⁸.)

Wait — actually, in many school worksheets, they prefer positive exponents. Let me double-check standard practice. Since Problem 1 had a⁻¹ and we combined it, and didn’t convert to fraction, perhaps we can leave negative exponents. But to be safe, I’ll write both forms and pick the most common.

Actually, let’s look at Problem 3 — it will likely result in positive exponents. For now, I’ll write 2a⁻⁸, but note that 2/a⁸ is also correct. However, since the original problem used negative exponent (a⁻¹), maybe they allow it. I’ll stick with 2a⁻⁸ for consistency with input format.

But wait — let’s see what happens in Problem 5 — it has s⁻¹ and (s²)⁻⁴, so probably they expect answers with negative exponents allowed. So I’ll keep 2a⁻⁸.

Actually, rethinking: In algebra, “simplify” often means write with positive exponents unless specified. Let me check online standards — but since this is a worksheet from Math Monks, and their examples usually convert to positive exponents. To be safe, I’ll convert final answers to positive exponents where possible.

So for Problem 2: 2a⁻⁸ = 2/a⁸

I think that’s better. Let’s do that for all.

Revised Plan: Final answers should have positive exponents unless the variable is in denominator already or something. Actually, let’s just simplify fully and use positive exponents.

For Problem 1: 2a⁸ — already positive.

Problem 2: 2a⁻⁸ → 2/a⁸

Yes.

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Problem 3: (x³y / xy⁵) · (x²y⁹ / x⁸)

First, simplify each fraction separately.

Left fraction: x³y / xy⁵
= x^(3-1) y^(1-5) = x² y⁻⁴

Right fraction: x²y⁹ / x⁸ = x^(2-8) y⁹ = x⁻⁶ y⁹

Now multiply them: (x² y⁻⁴) · (x⁻⁶ y⁹)
= x^(2 + -6) y^(-4 + 9) = x⁻⁴ y⁵

Convert to positive exponents: y⁵ / x⁴

Final: y⁵ / x⁴

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Problem 4: (8m⁵n⁷ / 2mn⁵)³

First, simplify inside the parentheses.

8/2 = 4
m⁵/m = m⁴
n⁷/n⁵ = n²

So inside: 4m⁴n²

Now raise to third power: (4m⁴n²)³ = 4³ · (m⁴)³ · (n²)³ = 64 m¹² n⁶

Final: 64m¹²n⁶

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Problem 5: [3⁻² s⁵ (s⁻¹)⁻³] / [s (s²)⁻⁴]

First, simplify numerator and denominator separately.

Numerator: 3⁻² s⁵ (s⁻¹)⁻³
(s⁻¹)⁻³ = s^[(-1)*(-3)] = s³
So numerator: 3⁻² s⁵ s³ = 3⁻² s^(5+3) = 3⁻² s⁸

Denominator: s (s²)⁻⁴
(s²)⁻⁴ = s^[2*(-4)] = s⁻⁸
So denominator: s · s⁻⁸ = s^(1 + -8) = s⁻⁷

Now overall: (3⁻² s⁸) / (s⁻⁷) = 3⁻² s^(8 - (-7)) = 3⁻² s^(15)

3⁻² = 1/9, so: (1/9) s¹⁵ = s¹⁵ / 9

Final: s¹⁵ / 9

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Problem 6: (3x³y⁻¹z⁻¹) / (x⁴y⁰z⁰)

Note: y⁰ = 1, z⁰ = 1, so denominator is x⁴ * 1 * 1 = x⁴

Numerator: 3x³y⁻¹z⁻¹

So: 3x³y⁻¹z⁻¹ / x⁴ = 3 x^(3-4) y⁻¹ z⁻¹ = 3 x⁻¹ y⁻¹ z⁻¹

Convert to positive exponents: 3 / (x y z)

Final: 3/(xyz)

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Problem 7: (x⁸ / 2y) · (5y² / x³)

Multiply numerators and denominators:

Numerator: x⁸ · 5y² = 5 x⁸ y²
Denominator: 2y · x³ = 2 x³ y

So: (5 x⁸ y²) / (2 x³ y) = (5/2) x^(8-3) y^(2-1) = (5/2) x⁵ y¹ = (5/2) x⁵ y

Final: (5/2)x⁵y or 5x⁵y/2

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Problem 8: (2x²y⁴ · 4x²y⁴ · 3x) / (3x⁻³y²)

First, numerator: multiply all together.

Coefficients: 2 · 4 · 3 = 24
x terms: x² · x² · x = x^(2+2+1) = x⁵
y terms: y⁴ · y⁴ = y⁸

So numerator: 24 x⁵ y⁸

Denominator: 3 x⁻³ y²

Now divide: (24 x⁵ y⁸) / (3 x⁻³ y²) = (24/3) x^(5 - (-3)) y^(8-2) = 8 x^(8) y⁶

Final: 8x⁸y⁶

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Now, compiling all answers:

1. 2a⁸
2. 2/a⁸
3. y⁵/x⁴
4. 64m¹²n⁶
5. s¹⁵/9
6. 3/(xyz)
7. 5x⁵y/2
8. 8x⁸y⁶

Double-checked calculations — all seem correct.

Final Answer:
1. \( 2a^8 \)
2. \( \frac{2}{a^8} \)
3. \( \frac{y^5}{x^4} \)
4. \( 64m^{12}n^6 \)
5. \( \frac{s^{15}}{9} \)
6. \( \frac{3}{xyz} \)
7. \( \frac{5x^5y}{2} \)
8. \( 8x^8y^6 \)