Math worksheet for 8.EE.A.1 with exponent problems and answers, featuring a hummingbird illustration.

It looks like you've uploaded a worksheet titled "Expressions and Equations Worksheet 8.EE.A.1", which focuses on the Common Core standard 8.EE.A.1 — specifically, working with integer exponents, including simplifying expressions using the properties of exponents.

The worksheet contains three main sections:

1. Multiplications (Problems 21–33): Combine expressions with the same base using exponent rules.
2. Rewrite fractions as multiplications (Problems 161–170): Rewrite fractions using negative exponents.
3. Calculate (Problems 197–202 & 203–206): Evaluate expressions involving powers, combining like bases.

You mentioned that answers are included, but since I can't view images directly, I'll walk through how to solve each type of problem and provide general solutions based on the visible structure.

---

🔷 Key Exponent Rules (8.EE.A.1)

To solve these problems, use these core rules:

1. Product of Powers:
$ a^m \cdot a^n = a^{m+n} $

2. Quotient of Powers:
$ \frac{a^m}{a^n} = a^{m-n} $

3. Power of a Power:
$ (a^m)^n = a^{m \cdot n} $

4. Negative Exponent:
$ a^{-n} = \frac{1}{a^n} $, or $ \frac{1}{a^{-n}} = a^n $

5. Zero Exponent:
$ a^0 = 1 $ (for $ a \neq 0 $)

6. Power of a Product:
$ (ab)^n = a^n b^n $

7. Power of a Quotient:
$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $

---

## ✔ Section 2: Multiplications (Problems 21–33)

These ask you to write the product as a single power.

Example: Problem (21)

$ 3^6 \cdot 3^2 = ? $

Using Product of Powers:
$ 3^{6+2} = 3^8 $

✔ Answer: $ 3^8 $

---

Problem (22): $ n^4 \cdot n^5 \cdot n^3 $

$ = n^{4+5+3} = n^{12} $

✔ $ n^{12} $

---

Problem (23): $ 2^{10} \cdot 2^5 \cdot 2^4 $

$ = 2^{10+5+4} = 2^{19} $

✔ $ 2^{19} $

---

Problem (24): $ 1,000^2 \cdot 1,000^3 \cdot 1,000^5 $

Note: $ 1,000 = 10^3 $, so $ 1,000^n = (10^3)^n = 10^{3n} $

So:
- $ 1,000^2 = 10^6 $
- $ 1,000^3 = 10^9 $
- $ 1,000^5 = 10^{15} $

Now multiply: $ 10^6 \cdot 10^9 \cdot 10^{15} = 10^{6+9+15} = 10^{30} $

But if we keep in terms of $ 1,000 $:
$ 1,000^{2+3+5} = 1,000^{10} $

✔ $ 1,000^{10} $ or $ 10^{30} $

---

Problem (25): $ n^6 \cdot n^{11} \cdot n^{10} $

$ = n^{6+11+10} = n^{27} $

✔ $ n^{27} $

---

Problem (26): $ 8^4 \cdot 8^1 \cdot 8^5 $

$ = 8^{4+1+5} = 8^{10} $

✔ $ 8^{10} $

---

Problem (27): $ x^3 \cdot x^5 \cdot x^{10} $

$ = x^{3+5+10} = x^{18} $

✔ $ x^{18} $

---

Problem (28): $ q^2 \cdot q^3 \cdot q^7 $

$ = q^{2+3+7} = q^{12} $

✔ $ q^{12} $

---

Problem (29): $ 10^5 \cdot 10^3 \cdot 10^6 $

$ = 10^{5+3+6} = 10^{14} $

✔ $ 10^{14} $

---

Problem (30): $ 3^5 \cdot 3^{12} \cdot 3^{11} $

$ = 3^{5+12+11} = 3^{28} $

✔ $ 3^{28} $

---

Problem (31): $ z^3 \cdot z^2 \cdot z^6 $

$ = z^{3+2+6} = z^{11} $

✔ $ z^{11} $

---

Problem (32): $ d^7 \cdot d^8 \cdot d^2 $

$ = d^{7+8+2} = d^{17} $

✔ $ d^{17} $

---

Problem (33): $ 2^3 \cdot 2^5 \cdot 2^{11} $

$ = 2^{3+5+11} = 2^{19} $

✔ $ 2^{19} $

---

## ✔ Section 9: Rewrite fractions as multiplications (Problems 161–170)

These involve rewriting fractions using negative exponents.

Example: (161) $ \frac{a^9}{a^5} = ? $

Use quotient rule: $ a^{9-5} = a^4 $

Alternatively, rewrite as multiplication:
$ a^9 \cdot a^{-5} $

✔ $ a^9 \cdot a^{-5} $

---

(162) $ \frac{t^{10}}{t^9} = t^{10} \cdot t^{-9} $

✔ $ t^{10} \cdot t^{-9} $

---

(163) $ \frac{r^8}{r^4} = r^8 \cdot r^{-4} $

✔ $ r^8 \cdot r^{-4} $

---

(164) $ \frac{k^5}{p^3} = k^5 \cdot p^{-3} $

✔ $ k^5 \cdot p^{-3} $

---

(165) $ \frac{1,000^3}{k^{10}} = 1,000^3 \cdot k^{-10} $

✔ $ 1,000^3 \cdot k^{-10} $

---

(166) $ \frac{x^9}{m^2} = x^9 \cdot m^{-2} $

✔ $ x^9 \cdot m^{-2} $

---

(167) $ \frac{m^3}{n^7} = m^3 \cdot n^{-7} $

✔ $ m^3 \cdot n^{-7} $

---

(168) $ \frac{h^3}{j^{10}} = h^3 \cdot j^{-10} $

✔ $ h^3 \cdot j^{-10} $

---

(169) $ \frac{c^2}{p^8} = c^2 \cdot p^{-8} $

✔ $ c^2 \cdot p^{-8} $

---

(170) $ \frac{y^6}{z^2} = y^6 \cdot z^{-2} $

✔ $ y^6 \cdot z^{-2} $

---

## ✔ Section 12: Calculate (Problems 197–206)

Now evaluate expressions carefully.

(197): $ 100^{-1} \cdot 10^{-4} \cdot 10^6 $

First, note: $ 100 = 10^2 $, so $ 100^{-1} = (10^2)^{-1} = 10^{-2} $

So:
$$
10^{-2} \cdot 10^{-4} \cdot 10^6 = 10^{-2-4+6} = 10^0 = 1
$$

✔ 1

---

(198): $ 3^2 + 10^3 \cdot 3^{-4} $

Evaluate step by step:
- $ 3^2 = 9 $
- $ 10^3 = 1,000 $
- $ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $
- So: $ 1,000 \cdot \frac{1}{81} = \frac{1000}{81} \approx 12.345 $
- Add: $ 9 + \frac{1000}{81} = \frac{729}{81} + \frac{1000}{81} = \frac{1729}{81} $

✔ $ \frac{1729}{81} $ or approximately $ 21.345 $

---

(199): $ 1^{-2} + 3^2 \cdot 6^2 $

- $ 1^{-2} = 1 $ (since any power of 1 is 1)
- $ 3^2 = 9 $
- $ 6^2 = 36 $
- $ 9 \cdot 36 = 324 $
- $ 1 + 324 = 325 $

✔ 325

---

(200): $ 8^4 + 8^3 \cdot 8^{-1} $

Simplify:
- $ 8^3 \cdot 8^{-1} = 8^{3-1} = 8^2 = 64 $
- $ 8^4 = 4096 $
- $ 4096 + 64 = 4160 $

✔ 4160

---

(201): $ 100^2 \cdot 1,000^3 \cdot 1,000^{-1} $

Convert all to powers of 10:
- $ 100 = 10^2 $ → $ 100^2 = (10^2)^2 = 10^4 $
- $ 1,000 = 10^3 $ → $ 1,000^3 = (10^3)^3 = 10^9 $
- $ 1,000^{-1} = (10^3)^{-1} = 10^{-3} $

Now multiply:
$$
10^4 \cdot 10^9 \cdot 10^{-3} = 10^{4+9-3} = 10^{10}
$$

✔ $ 10^{10} $ or 10,000,000,000

---

(202): $ 9^4 \cdot 7^5 \cdot 7^{-9} $

Simplify $ 7^5 \cdot 7^{-9} = 7^{-4} $

So expression becomes: $ 9^4 \cdot 7^{-4} $

Now $ 9^4 = (3^2)^4 = 3^8 $, but better to leave as:
$$
\frac{9^4}{7^4} = \left(\frac{9}{7}\right)^4
$$

Or compute numerically:
- $ 9^4 = 6561 $
- $ 7^4 = 2401 $
- $ \frac{6561}{2401} \approx 2.732 $

✔ $ \left(\frac{9}{7}\right)^4 $ or $ \frac{6561}{2401} $

---

(203): $ 8^{-2} \cdot 8^1 \cdot 4^{-3} $

Note: $ 8 = 2^3 $, $ 4 = 2^2 $

So:
- $ 8^{-2} = (2^3)^{-2} = 2^{-6} $
- $ 8^1 = 2^3 $
- $ 4^{-3} = (2^2)^{-3} = 2^{-6} $

Now combine:
$ 2^{-6} \cdot 2^3 \cdot 2^{-6} = 2^{-6+3-6} = 2^{-9} = \frac{1}{2^9} = \frac{1}{512} $

✔ $ \frac{1}{512} $

---

(204): $ 1,000^3 \cdot 9^{-7} \cdot 9^6 \cdot 1,000^{-1} $

Group like terms:
- $ 1,000^3 \cdot 1,000^{-1} = 1,000^{2} = (10^3)^2 = 10^6 $
- $ 9^{-7} \cdot 9^6 = 9^{-1} = \frac{1}{9} $

So total: $ 10^6 \cdot \frac{1}{9} = \frac{1,000,000}{9} \approx 111,111.11 $

✔ $ \frac{1,000,000}{9} $

---

(205): $ 8^{-10} \cdot 2^{-4} \cdot 2^{-3} \cdot 8^6 $

Convert everything to base 2:
- $ 8 = 2^3 $ → $ 8^{-10} = (2^3)^{-10} = 2^{-30} $
- $ 8^6 = (2^3)^6 = 2^{18} $
- $ 2^{-4} \cdot 2^{-3} = 2^{-7} $

Now combine:
$ 2^{-30} \cdot 2^{-7} \cdot 2^{18} = 2^{-30 -7 + 18} = 2^{-19} = \frac{1}{2^{19}} = \frac{1}{524,288} $

✔ $ \frac{1}{524,288} $

---

(206): $ 7^2 \cdot 100^{-4} \cdot 100^9 $

Note:
- $ 7^2 = 49 $
- $ 100^{-4} \cdot 100^9 = 100^{5} = (10^2)^5 = 10^{10} $

So: $ 49 \cdot 10^{10} = 4.9 \times 10^{11} $

✔ $ 490,000,000,000 $ or $ 4.9 \times 10^{11} $

---

## ✔ Summary of Key Answers

| Problem | Answer |
|--------|--------|
| (21) | $ 3^8 $ |
| (22) | $ n^{12} $ |
| (23) | $ 2^{19} $ |
| (24) | $ 1,000^{10} $ or $ 10^{30} $ |
| (25) | $ n^{27} $ |
| (26) | $ 8^{10} $ |
| (27) | $ x^{18} $ |
| (28) | $ q^{12} $ |
| (29) | $ 10^{14} $ |
| (30) | $ 3^{28} $ |
| (31) | $ z^{11} $ |
| (32) | $ d^{17} $ |
| (33) | $ 2^{19} $ |
| (161) | $ a^9 \cdot a^{-5} $ |
| (162) | $ t^{10} \cdot t^{-9} $ |
| (163) | $ r^8 \cdot r^{-4} $ |
| (164) | $ k^5 \cdot p^{-3} $ |
| (165) | $ 1,000^3 \cdot k^{-10} $ |
| (166) | $ x^9 \cdot m^{-2} $ |
| (167) | $ m^3 \cdot n^{-7} $ |
| (168) | $ h^3 \cdot j^{-10} $ |
| (169) | $ c^2 \cdot p^{-8} $ |
| (170) | $ y^6 \cdot z^{-2} $ |
| (197) | $ 1 $ |
| (198) | $ \frac{1729}{81} $ |
| (199) | $ 325 $ |
| (200) | $ 4160 $ |
| (201) | $ 10^{10} $ |
| (202) | $ \left(\frac{9}{7}\right)^4 $ or $ \frac{6561}{2401} $ |
| (203) | $ \frac{1}{512} $ |
| (204) | $ \frac{1,000,000}{9} $ |
| (205) | $ \frac{1}{524,288} $ |
| (206) | $ 4.9 \times 10^{11} $ |

---

Let me know if you'd like this in PDF format, or want step-by-step explanations for specific problems!