Exponents and Radicals Worksheets | Exponents & Radicals ... - Free Printable

Problem: Simplify the given expressions. Your answer should contain only positive exponents.

We will solve each problem step by step, using the rules of exponents:

#### Rule Recap:
1. Division of Exponents: $\frac{a^m}{a^n} = a^{m-n}$
2. Multiplication of Exponents: $a^m \cdot a^n = a^{m+n}$
3. Negative Exponents: $a^{-n} = \frac{1}{a^n}$

---

Problem 1:

$$
\frac{h}{h^2}
$$
- Using the division rule: $\frac{h}{h^2} = h^{1-2} = h^{-1}$
- Convert to positive exponent: $h^{-1} = \frac{1}{h}$

Answer:
$$
\boxed{\frac{1}{h}}
$$

---

Problem 2:

$$
\frac{d^4}{d^x}
$$
- Using the division rule: $\frac{d^4}{d^x} = d^{4-x}$

Answer:
$$
\boxed{d^{4-x}}
$$

---

Problem 3:

$$
c^5 \cdot c^6 \cdot c^4
$$
- Using the multiplication rule: $c^5 \cdot c^6 \cdot c^4 = c^{5+6+4} = c^{15}$

Answer:
$$
\boxed{c^{15}}
$$

---

Problem 4:

$$
\frac{6b^2g^3}{8b^2g^4}
$$
- Simplify coefficients: $\frac{6}{8} = \frac{3}{4}$
- Simplify variables:
- For $b$: $\frac{b^2}{b^2} = b^{2-2} = b^0 = 1$
- For $g$: $\frac{g^3}{g^4} = g^{3-4} = g^{-1} = \frac{1}{g}$
- Combine: $\frac{3}{4} \cdot 1 \cdot \frac{1}{g} = \frac{3}{4g}$

Answer:
$$
\boxed{\frac{3}{4g}}
$$

---

Problem 5:

$$
\frac{2b^3n^2}{5bn^2}
$$
- Simplify coefficients: $\frac{2}{5}$
- Simplify variables:
- For $b$: $\frac{b^3}{b} = b^{3-1} = b^2$
- For $n$: $\frac{n^2}{n^2} = n^{2-2} = n^0 = 1$
- Combine: $\frac{2}{5} \cdot b^2 \cdot 1 = \frac{2b^2}{5}$

Answer:
$$
\boxed{\frac{2b^2}{5}}
$$

---

Problem 6:

$$
\frac{9zd^3}{4z^2d^2}
$$
- Simplify coefficients: $\frac{9}{4}$
- Simplify variables:
- For $z$: $\frac{z}{z^2} = z^{1-2} = z^{-1} = \frac{1}{z}$
- For $d$: $\frac{d^3}{d^2} = d^{3-2} = d^1 = d$
- Combine: $\frac{9}{4} \cdot \frac{1}{z} \cdot d = \frac{9d}{4z}$

Answer:
$$
\boxed{\frac{9d}{4z}}
$$

---

Problem 7:

$$
2d \cdot 4d^2
$$
- Multiply coefficients: $2 \cdot 4 = 8$
- Multiply variables: $d \cdot d^2 = d^{1+2} = d^3$
- Combine: $8d^3$

Answer:
$$
\boxed{8d^3}
$$

---

Problem 8:

$$
5k^3 \cdot 9k^2
$$
- Multiply coefficients: $5 \cdot 9 = 45$
- Multiply variables: $k^3 \cdot k^2 = k^{3+2} = k^5$
- Combine: $45k^5$

Answer:
$$
\boxed{45k^5}
$$

---

Problem 9:

$$
\frac{k^x}{k^{-3}}
$$
- Using the division rule: $\frac{k^x}{k^{-3}} = k^{x - (-3)} = k^{x+3}$

Answer:
$$
\boxed{k^{x+3}}
$$

---

Problem 10:

$$
z^3w^{-6} \cdot 8z^{-6}w^4 \cdot 4zw^5
$$
- Multiply coefficients: $1 \cdot 8 \cdot 4 = 32$
- Multiply variables:
- For $z$: $z^3 \cdot z^{-6} \cdot z^1 = z^{3-6+1} = z^{-2} = \frac{1}{z^2}$
- For $w$: $w^{-6} \cdot w^4 \cdot w^5 = w^{-6+4+5} = w^3$
- Combine: $32 \cdot \frac{1}{z^2} \cdot w^3 = \frac{32w^3}{z^2}$

Answer:
$$
\boxed{\frac{32w^3}{z^2}}
$$

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Problem 11:

$$
w^3 \cdot 6w^6b^4
$$
- Multiply coefficients: $1 \cdot 6 = 6$
- Multiply variables:
- For $w$: $w^3 \cdot w^6 = w^{3+6} = w^9$
- For $b$: $b^4$ remains as is
- Combine: $6w^9b^4$

Answer:
$$
\boxed{6w^9b^4}
$$

---

Problem 12:

$$
4c^3 \cdot 5c^4
$$
- Multiply coefficients: $4 \cdot 5 = 20$
- Multiply variables: $c^3 \cdot c^4 = c^{3+4} = c^7$
- Combine: $20c^7$

Answer:
$$
\boxed{20c^7}
$$

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Final Answers:

1. $\boxed{\frac{1}{h}}$
2. $\boxed{d^{4-x}}$
3. $\boxed{c^{15}}$
4. $\boxed{\frac{3}{4g}}$
5. $\boxed{\frac{2b^2}{5}}$
6. $\boxed{\frac{9d}{4z}}$
7. $\boxed{8d^3}$
8. $\boxed{45k^5}$
9. $\boxed{k^{x+3}}$
10. $\boxed{\frac{32w^3}{z^2}}$
11. $\boxed{6w^9b^4}$
12. $\boxed{20c^7}$