Simplify integer exponent problems from Worksheet #20 Section 5.1, focusing on expressions with negative exponents.

Let's solve each problem from Worksheet #20 Section 5.1 — Integer Exponent Problems step by step, simplifying expressions with integer exponents and ensuring no negative exponents in the final answer.

We'll use the following exponent rules:

1. $ a^{-n} = \frac{1}{a^n} $
2. $ a^m \cdot a^n = a^{m+n} $
3. $ \frac{a^m}{a^n} = a^{m-n} $
4. $ (a^m)^n = a^{m \cdot n} $
5. $ (ab)^n = a^n b^n $
6. $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $

---

(1) $ 5^2 $

$$
= 25
$$

---

(2) $ x^{-3} $

$$
= \frac{1}{x^3}
$$

---

(3) $ x^{-4} x^2 $

$$
= x^{-4+2} = x^{-2} = \frac{1}{x^2}
$$

---

(4) $ -2x^{-3} $

$$
= -\frac{2}{x^3}
$$

---

(5) $ x^3 $

$$
= x^3 \quad \text{(already simplified)}
$$

---

(6) $ x^2 x^3 $

$$
= x^{2+3} = x^5
$$

---

(7) $ \left(\frac{2}{3}\right)^{-1} $

$$
= \frac{3}{2}
$$

---

(8) $ \left(\frac{2}{3}\right)^{-3} $

$$
= \left(\frac{3}{2}\right)^3 = \frac{27}{8}
$$

---

(9) $ (2x^3)(5x^2) $

$$
= 2 \cdot 5 \cdot x^{3+2} = 10x^5
$$

---

(10) $ (-3x^2)(5x^7) $

$$
= -15x^{2+7} = -15x^9
$$

---

(11) $ 3x^{-6} x^5 $

$$
= 3x^{-6+5} = 3x^{-1} = \frac{3}{x}
$$

---

(12) $ (2xy)(5x^2y^4) $

$$
= 2 \cdot 5 \cdot x^{1+2} y^{1+4} = 10x^3 y^5
$$

---

(13) $ -4x^2 y (2x^4 y^6) $

$$
= -8x^{2+4} y^{1+6} = -8x^6 y^7
$$

---

(14) $ x^{-2} $

$$
= \frac{1}{x^2}
$$

---

(15) $ 2^{-3} $

$$
= \frac{1}{2^3} = \frac{1}{8}
$$

---

(16) $ \left(\frac{2}{3}\right)^3 $

$$
= \frac{8}{27}
$$

---

(17) $ \left(\frac{2}{3}\right)^{-2} $

$$
= \left(\frac{3}{2}\right)^2 = \frac{9}{4}
$$

---

(18) $ \left(\frac{x}{y}\right)^2 $

$$
= \frac{x^2}{y^2}
$$

---

(19) $ (x^2)^3 $

$$
= x^{2 \cdot 3} = x^6
$$

---

(20) $ 3^2 x^2 $

$$
= 9x^2
$$

---

(21) $ \frac{x^{-4}}{y^{-3}} $

$$
= \frac{y^3}{x^4}
$$

---

(22) $ \left(\frac{1}{2}\right)^{-1} $

$$
= 2
$$

---

(23) $ \frac{1}{x^{-2}} $

$$
= x^2
$$

---

(24) $ (x^3)^{-2} $

$$
= x^{-6} = \frac{1}{x^6}
$$

---

(25) $ (x^{-3})^{-1} $

$$
= x^{(-3)(-1)} = x^3
$$

---

(26) $ 6^{-1} $

$$
= \frac{1}{6}
$$

---

(27) $ 2x^4 (x^6) $

$$
= 2x^{4+6} = 2x^{10}
$$

---

(28) $ (2x^3)^2 $

$$
= 2^2 x^{3 \cdot 2} = 4x^6
$$

---

(29) $ \left(\frac{x}{y}\right)^{-3} $

$$
= \left(\frac{y}{x}\right)^3 = \frac{y^3}{x^3}
$$

---

(30) $ (2y)^2 $

$$
= 4y^2
$$

---

(31) $ \frac{x^5}{x^2} $

$$
= x^{5-2} = x^3
$$

---

(32) $ 2^{-1} $

$$
= \frac{1}{2}
$$

---

(33) $ \frac{z^2}{z^{-2}} $

$$
= z^{2 - (-2)} = z^4
$$

---

(34) $ \frac{z^2}{z^{-3}} $

$$
= z^{2 - (-3)} = z^5
$$

---

(35) $ (-4)^{-2} $

$$
= \frac{1}{(-4)^2} = \frac{1}{16}
$$

---

(36) $ \frac{-26^{-2}}{} $

Wait: This is written as $ \frac{-26^{-2}}{} $? Likely typo.

Assuming it’s just $ -26^{-2} $:
$$
= -\frac{1}{26^2} = -\frac{1}{676}
$$

---

(37) $ (x^2 y^3)^{-1} $

$$
= \frac{1}{x^2 y^3}
$$

---

(38) $ -3^2 $

Note: Order of operations — exponent before negation.
$$
= -(3^2) = -9
$$

---

(39) $ \frac{2^3}{2^5} $

$$
= 2^{3-5} = 2^{-2} = \frac{1}{4}
$$

---

(40) $ 2x^{-2} $

$$
= \frac{2}{x^2}
$$

---

(41) $ \frac{1}{x^{-2}} $

$$
= x^2
$$

---

(42) $ \frac{1}{x^{-2}} $

Same as (41): $ = x^2 $

Wait, but (41) and (42) are both $ \frac{1}{x^{-2}} $? Possibly a repeat or typo.

But let's assume (42) is $ \frac{1}{x^{-2}} $ → $ x^2 $

---

(43) $ 3^{-2} $

$$
= \frac{1}{3^2} = \frac{1}{9}
$$

---

(44) $ \frac{x^2}{x^{-1}} (2xy)^2 (3x^2)^{-3} $

Break this down:

First term: $ \frac{x^2}{x^{-1}} = x^{2 - (-1)} = x^3 $

Second: $ (2xy)^2 = 4x^2 y^2 $

Third: $ (3x^2)^{-3} = 3^{-3} x^{-6} = \frac{1}{27} x^{-6} $

Now multiply all:
$$
x^3 \cdot 4x^2 y^2 \cdot \frac{1}{27} x^{-6}
= \frac{4}{27} x^{3+2-6} y^2 = \frac{4}{27} x^{-1} y^2 = \frac{4y^2}{27x}
$$

---

(45) $ (-3)^2 $

$$
= 9
$$

---

(46) $ (-2)^{-3} $

$$
= \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}
$$

---

(47) $ 0^0 $

This is undefined or sometimes considered indeterminate. In many contexts, it's left undefined.

But if we follow convention for algebra problems, often $ 0^0 = 1 $ is assumed.

But strictly speaking, undefined.

However, since the worksheet asks to simplify, and assuming they want an answer, perhaps write:
$$
\text{Undefined}
$$

But some textbooks define $ 0^0 = 1 $. We’ll note it as undefined unless specified.

---

(48) $ \frac{x^{-1}}{y^{-2}} $

$$
= \frac{y^2}{x}
$$

---

(49) $ 8^{-1} $

$$
= \frac{1}{8}
$$

---

(50) $ x^{-5} $

$$
= \frac{1}{x^5}
$$

---

(51) $ x^{-2} $

$$
= \frac{1}{x^2}
$$

---

(52) $ 8^{-2} $

$$
= \frac{1}{8^2} = \frac{1}{64}
$$

---

(53) $ x^3 x^{-2} $

$$
= x^{3-2} = x^1 = x
$$

---

(54) $ (x^{-2})^{-2} $

$$
= x^{(-2)(-2)} = x^4
$$

---

(55) $ (x^3)^{-2} $

$$
= x^{-6} = \frac{1}{x^6}
$$

---

(56) $ \frac{x^2 x^4}{x^3} $

$$
= \frac{x^{6}}{x^3} = x^{3}
$$

---

(57) $ (2x^{-2})^2 $

$$
= 2^2 x^{-4} = 4x^{-4} = \frac{4}{x^4}
$$

---

## ✔ Final Answers (Cleaned Up):

| Problem | Answer |
|--------|--------|
| (1) | $ 25 $ |
| (2) | $ \frac{1}{x^3} $ |
| (3) | $ \frac{1}{x^2} $ |
| (4) | $ -\frac{2}{x^3} $ |
| (5) | $ x^3 $ |
| (6) | $ x^5 $ |
| (7) | $ \frac{3}{2} $ |
| (8) | $ \frac{27}{8} $ |
| (9) | $ 10x^5 $ |
| (10) | $ -15x^9 $ |
| (11) | $ \frac{3}{x} $ |
| (12) | $ 10x^3 y^5 $ |
| (13) | $ -8x^6 y^7 $ |
| (14) | $ \frac{1}{x^2} $ |
| (15) | $ \frac{1}{8} $ |
| (16) | $ \frac{8}{27} $ |
| (17) | $ \frac{9}{4} $ |
| (18) | $ \frac{x^2}{y^2} $ |
| (19) | $ x^6 $ |
| (20) | $ 9x^2 $ |
| (21) | $ \frac{y^3}{x^4} $ |
| (22) | $ 2 $ |
| (23) | $ x^2 $ |
| (24) | $ \frac{1}{x^6} $ |
| (25) | $ x^3 $ |
| (26) | $ \frac{1}{6} $ |
| (27) | $ 2x^{10} $ |
| (28) | $ 4x^6 $ |
| (29) | $ \frac{y^3}{x^3} $ |
| (30) | $ 4y^2 $ |
| (31) | $ x^3 $ |
| (32) | $ \frac{1}{2} $ |
| (33) | $ z^4 $ |
| (34) | $ z^5 $ |
| (35) | $ \frac{1}{16} $ |
| (36) | $ -\frac{1}{676} $ |
| (37) | $ \frac{1}{x^2 y^3} $ |
| (38) | $ -9 $ |
| (39) | $ \frac{1}{4} $ |
| (40) | $ \frac{2}{x^2} $ |
| (41) | $ x^2 $ |
| (42) | $ x^2 $ |
| (43) | $ \frac{1}{9} $ |
| (44) | $ \frac{4y^2}{27x} $ |
| (45) | $ 9 $ |
| (46) | $ -\frac{1}{8} $ |
| (47) | $ \text{undefined} $ |
| (48) | $ \frac{y^2}{x} $ |
| (49) | $ \frac{1}{8} $ |
| (50) | $ \frac{1}{x^5} $ |
| (51) | $ \frac{1}{x^2} $ |
| (52) | $ \frac{1}{64} $ |
| (53) | $ x $ |
| (54) | $ x^4 $ |
| (55) | $ \frac{1}{x^6} $ |
| (56) | $ x^3 $ |
| (57) | $ \frac{4}{x^4} $ |

---

✔ All answers have been simplified with no negative exponents, except where the expression itself evaluates to a constant or fraction.

Let me know if you'd like these formatted in a printable sheet!