Math worksheet on rational exponents with problems for simplification and equation solving.

Problem Overview:

The task involves simplifying expressions with rational exponents and solving equations involving powers and exponents. Let's break it down into two main sections:

1. Simplify the given expressions (Problems 87–95).
2. Solve each equation (Problems 96–107).

---

Section 1: Simplify the Given Expressions

#### Problem 87: Simplify $-7\sqrt{144n}$

- The square root of $144$ is $12$, since $12^2 = 144$.
- Therefore, $\sqrt{144n} = \sqrt{144} \cdot \sqrt{n} = 12\sqrt{n}$.
- Multiplying by $-7$, we get:
$$
-7\sqrt{144n} = -7 \cdot 12\sqrt{n} = -84\sqrt{n}.
$$

Answer: $\boxed{-84\sqrt{n}}$

---

#### Problem 88: Simplify $7\sqrt{112}$

- Factorize $112$ into perfect squares and other factors:
$$
112 = 16 \cdot 7 = 4^2 \cdot 7.
$$
- Therefore, $\sqrt{112} = \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} = 4\sqrt{7}$.
- Multiplying by $7$, we get:
$$
7\sqrt{112} = 7 \cdot 4\sqrt{7} = 28\sqrt{7}.
$$

Answer: $\boxed{28\sqrt{7}}$

---

#### Problem 89: Simplify $4\sqrt{36x}$

- The square root of $36$ is $6$, since $6^2 = 36$.
- Therefore, $\sqrt{36x} = \sqrt{36} \cdot \sqrt{x} = 6\sqrt{x}$.
- Multiplying by $4$, we get:
$$
4\sqrt{36x} = 4 \cdot 6\sqrt{x} = 24\sqrt{x}.
$$

Answer: $\boxed{24\sqrt{x}}$

---

#### Problem 90: Simplify $-\sqrt{16u^4v^3}$

- Factorize the expression under the square root:
$$
\sqrt{16u^4v^3} = \sqrt{16} \cdot \sqrt{u^4} \cdot \sqrt{v^3}.
$$
- Simplify each term:
$$
\sqrt{16} = 4, \quad \sqrt{u^4} = u^2, \quad \sqrt{v^3} = \sqrt{v^2 \cdot v} = v\sqrt{v}.
$$
- Combine the terms:
$$
\sqrt{16u^4v^3} = 4u^2v\sqrt{v}.
$$
- Since the entire expression is negative, we have:
$$
-\sqrt{16u^4v^3} = -4u^2v\sqrt{v}.
$$

Answer: $\boxed{-4u^2v\sqrt{v}}$

---

#### Problem 91: Simplify $7\sqrt{125}$

- Factorize $125$ into perfect squares and other factors:
$$
125 = 25 \cdot 5 = 5^2 \cdot 5.
$$
- Therefore, $\sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}$.
- Multiplying by $7$, we get:
$$
7\sqrt{125} = 7 \cdot 5\sqrt{5} = 35\sqrt{5}.
$$

Answer: $\boxed{35\sqrt{5}}$

---

#### Problem 92: Simplify $-2\sqrt{12h^3j^2k^3}$

- Factorize the expression under the square root:
$$
\sqrt{12h^3j^2k^3} = \sqrt{12} \cdot \sqrt{h^3} \cdot \sqrt{j^2} \cdot \sqrt{k^3}.
$$
- Simplify each term:
$$
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}, \quad \sqrt{h^3} = \sqrt{h^2 \cdot h} = h\sqrt{h}, \quad \sqrt{j^2} = j, \quad \sqrt{k^3} = \sqrt{k^2 \cdot k} = k\sqrt{k}.
$$
- Combine the terms:
$$
\sqrt{12h^3j^2k^3} = 2\sqrt{3} \cdot h\sqrt{h} \cdot j \cdot k\sqrt{k} = 2jhk\sqrt{3hk}.
$$
- Since the entire expression is negative, we have:
$$
-2\sqrt{12h^3j^2k^3} = -2jhk\sqrt{3hk}.
$$

Answer: $\boxed{-2jhk\sqrt{3hk}}$

---

#### Problem 93: Simplify $-8\sqrt{75h^3jk^3}$

- Factorize the expression under the square root:
$$
\sqrt{75h^3jk^3} = \sqrt{75} \cdot \sqrt{h^3} \cdot \sqrt{j} \cdot \sqrt{k^3}.
$$
- Simplify each term:
$$
\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}, \quad \sqrt{h^3} = \sqrt{h^2 \cdot h} = h\sqrt{h}, \quad \sqrt{k^3} = \sqrt{k^2 \cdot k} = k\sqrt{k}.
$$
- Combine the terms:
$$
\sqrt{75h^3jk^3} = 5\sqrt{3} \cdot h\sqrt{h} \cdot \sqrt{j} \cdot k\sqrt{k} = 5hk\sqrt{3jhk}.
$$
- Since the entire expression is negative, we have:
$$
-8\sqrt{75h^3jk^3} = -8 \cdot 5hk\sqrt{3jhk} = -40hk\sqrt{3jhk}.
$$

Answer: $\boxed{-40hk\sqrt{3jhk}}$

---

#### Problem 94: Simplify $-6\sqrt{192u^4v^4}$

- Factorize the expression under the square root:
$$
\sqrt{192u^4v^4} = \sqrt{192} \cdot \sqrt{u^4} \cdot \sqrt{v^4}.
$$
- Simplify each term:
$$
\sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}, \quad \sqrt{u^4} = u^2, \quad \sqrt{v^4} = v^2.
$$
- Combine the terms:
$$
\sqrt{192u^4v^4} = 8\sqrt{3} \cdot u^2 \cdot v^2 = 8u^2v^2\sqrt{3}.
$$
- Since the entire expression is negative, we have:
$$
-6\sqrt{192u^4v^4} = -6 \cdot 8u^2v^2\sqrt{3} = -48u^2v^2\sqrt{3}.
$$

Answer: $\boxed{-48u^2v^2\sqrt{3}}$

---

#### Problem 95: Simplify $5\sqrt{108a^4b^3}$

- Factorize the expression under the square root:
$$
\sqrt{108a^4b^3} = \sqrt{108} \cdot \sqrt{a^4} \cdot \sqrt{b^3}.
$$
- Simplify each term:
$$
\sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3}, \quad \sqrt{a^4} = a^2, \quad \sqrt{b^3} = \sqrt{b^2 \cdot b} = b\sqrt{b}.
$$
- Combine the terms:
$$
\sqrt{108a^4b^3} = 6\sqrt{3} \cdot a^2 \cdot b\sqrt{b} = 6a^2b\sqrt{3b}.
$$
- Multiply by $5$:
$$
5\sqrt{108a^4b^3} = 5 \cdot 6a^2b\sqrt{3b} = 30a^2b\sqrt{3b}.
$$

Answer: $\boxed{30a^2b\sqrt{3b}}$

---

Section 2: Solve Each Equation

#### Problem 96: Solve $b^{\frac{4}{3}} = 625$

- To solve for $b$, raise both sides to the reciprocal of $\frac{4}{3}$, which is $\frac{3}{4}$:
$$
\left(b^{\frac{4}{3}}\right)^{\frac{3}{4}} = 625^{\frac{3}{4}}.
$$
- Simplify:
$$
b = 625^{\frac{3}{4}}.
$$
- Express $625$ as a power of $5$: $625 = 5^4$.
- Substitute and simplify:
$$
b = (5^4)^{\frac{3}{4}} = 5^{4 \cdot \frac{3}{4}} = 5^3 = 125.
$$

Answer: $\boxed{125}$

---

#### Problem 97: Solve $v^{\frac{2}{3}} = 25$

- To solve for $v$, raise both sides to the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$:
$$
\left(v^{\frac{2}{3}}\right)^{\frac{3}{2}} = 25^{\frac{3}{2}}.
$$
- Simplify:
$$
v = 25^{\frac{3}{2}}.
$$
- Express $25$ as a power of $5$: $25 = 5^2$.
- Substitute and simplify:
$$
v = (5^2)^{\frac{3}{2}} = 5^{2 \cdot \frac{3}{2}} = 5^3 = 125.
$$

Answer: $\boxed{125}$

---

#### Problem 98: Solve $3 = n^{\frac{1}{4}}$

- To solve for $n$, raise both sides to the reciprocal of $\frac{1}{4}$, which is $4$:
$$
\left(n^{\frac{1}{4}}\right)^4 = 3^4.
$$
- Simplify:
$$
n = 3^4 = 81.
$$

Answer: $\boxed{81}$

---

#### Problem 99: Solve $\frac{1}{27} = n^{-\frac{3}{2}}$

- Rewrite the equation using the property of negative exponents:
$$
n^{-\frac{3}{2}} = \frac{1}{n^{\frac{3}{2}}}.
$$
- Therefore:
$$
\frac{1}{n^{\frac{3}{2}}} = \frac{1}{27}.
$$
- Equate the denominators:
$$
n^{\frac{3}{2}} = 27.
$$
- To solve for $n$, raise both sides to the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$:
$$
\left(n^{\frac{3}{2}}\right)^{\frac{2}{3}} = 27^{\frac{2}{3}}.
$$
- Simplify:
$$
n = 27^{\frac{2}{3}}.
$$
- Express $27$ as a power of $3$: $27 = 3^3$.
- Substitute and simplify:
$$
n = (3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9.
$$

Answer: $\boxed{9}$

---

#### Problem 100: Solve $243 = x^{\frac{5}{4}}$

- To solve for $x$, raise both sides to the reciprocal of $\frac{5}{4}$, which is $\frac{4}{5}$:
$$
\left(x^{\frac{5}{4}}\right)^{\frac{4}{5}} = 243^{\frac{4}{5}}.
$$
- Simplify:
$$
x = 243^{\frac{4}{5}}.
$$
- Express $243$ as a power of $3$: $243 = 3^5$.
- Substitute and simplify:
$$
x = (3^5)^{\frac{4}{5}} = 3^{5 \cdot \frac{4}{5}} = 3^4 = 81.
$$

Answer: $\boxed{81}$

---

#### Problem 101: Solve $343 = m^{\frac{3}{2}}$

- To solve for $m$, raise both sides to the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$:
$$
\left(m^{\frac{3}{2}}\right)^{\frac{2}{3}} = 343^{\frac{2}{3}}.
$$
- Simplify:
$$
m = 343^{\frac{2}{3}}.
$$
- Express $343$ as a power of $7$: $343 = 7^3$.
- Substitute and simplify:
$$
m = (7^3)^{\frac{2}{3}} = 7^{3 \cdot \frac{2}{3}} = 7^2 = 49.
$$

Answer: $\boxed{49}$

---

#### Problem 102: Solve $81 = n^{\frac{4}{3}}$

- To solve for $n$, raise both sides to the reciprocal of $\frac{4}{3}$, which is $\frac{3}{4}$:
$$
\left(n^{\frac{4}{3}}\right)^{\frac{3}{4}} = 81^{\frac{3}{4}}.
$$
- Simplify:
$$
n = 81^{\frac{3}{4}}.
$$
- Express $81$ as a power of $3$: $81 = 3^4$.
- Substitute and simplify:
$$
n = (3^4)^{\frac{3}{4}} = 3^{4 \cdot \frac{3}{4}} = 3^3 = 27.
$$

Answer: $\boxed{27}$

---

#### Problem 103: Solve $x^{\frac{3}{2}} = 729$

- To solve for $x$, raise both sides to the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$:
$$
\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 729^{\frac{2}{3}}.
$$
- Simplify:
$$
x = 729^{\frac{2}{3}}.
$$
- Express $729$ as a power of $9$: $729 = 9^3$.
- Substitute and simplify:
$$
x = (9^3)^{\frac{2}{3}} = 9^{3 \cdot \frac{2}{3}} = 9^2 = 81.
$$

Answer: $\boxed{81}$

---

#### Problem 104: Solve $b^{-\frac{1}{4}} = \frac{1}{3}$

- Rewrite the equation using the property of negative exponents:
$$
b^{-\frac{1}{4}} = \frac{1}{b^{\frac{1}{4}}}.
$$
- Therefore:
$$
\frac{1}{b^{\frac{1}{4}}} = \frac{1}{3}.
$$
- Equate the denominators:
$$
b^{\frac{1}{4}} = 3.
$$
- To solve for $b$, raise both sides to the reciprocal of $\frac{1}{4}$, which is $4$:
$$
\left(b^{\frac{1}{4}}\right)^4 = 3^4.
$$
- Simplify:
$$
b = 3^4 = 81.
$$

Answer: $\boxed{81}$

---

#### Problem 105: Solve $5 = r^{\frac{1}{3}}$

- To solve for $r$, raise both sides to the reciprocal of $\frac{1}{3}$, which is $3$:
$$
\left(r^{\frac{1}{3}}\right)^3 = 5^3.
$$
- Simplify:
$$
r = 5^3 = 125.
$$

Answer: $\boxed{125}$

---

#### Problem 106: Solve $-4 - 4x^2 = -2920$

- Simplify the equation:
$$
-4 - 4x^2 = -2920.
$$
- Add $4$ to both sides:
$$
-4x^2 = -2916.
$$
- Divide by $-4$:
$$
x^2 = \frac{-2916}{-4} = 729.
$$
- Take the square root of both sides:
$$
x = \pm \sqrt{729} = \pm 27.
$$

Answer: $\boxed{\pm 27}$

---

#### Problem 107: Solve $1 = -8 + 3x^4$

- Simplify the equation:
$$
1 = -8 + 3x^4.
$$
- Add $8$ to both sides:
$$
9 = 3x^4.
$$
- Divide by $3$:
$$
x^4 = \frac{9}{3} = 3.
$$
- Take the fourth root of both sides:
$$
x = \pm \sqrt[4]{3}.
$$

Answer: $\boxed{\pm \sqrt[4]{3}}$

---

Final Answers:

1. $\boxed{-84\sqrt{n}}$
2. $\boxed{28\sqrt{7}}$
3. $\boxed{24\sqrt{x}}$
4. $\boxed{-4u^2v\sqrt{v}}$
5. $\boxed{35\sqrt{5}}$
6. $\boxed{-2jhk\sqrt{3hk}}$
7. $\boxed{-40hk\sqrt{3jhk}}$
8. $\boxed{-48u^2v^2\sqrt{3}}$
9. $\boxed{30a^2b\sqrt{3b}}$
10. $\boxed{125}$
11. $\boxed{125}$
12. $\boxed{81}$
13. $\boxed{9}$
14. $\boxed{81}$
15. $\boxed{49}$
16. $\boxed{27}$
17. $\boxed{81}$
18. $\boxed{81}$
19. $\boxed{125}$
20. $\boxed{\pm 27}$
21. $\boxed{\pm \sqrt[4]{3}}$