Worksheet on simplifying expressions with rational exponents and converting between radical and exponential forms.

Problem Analysis:

The task involves simplifying expressions involving roots (square roots, cube roots, etc.) and verifying whether certain statements are true or false. The goal is to simplify the given expressions and evaluate the truth of the provided statements.

Step-by-Step Solution:

#### Expressions to Simplify:

1. Expression: \( \sqrt{4} \)
- Simplification: \( \sqrt{4} = 2 \)
- Answer: \( 2 \)

2. Expression: \( \sqrt[3]{8} \)
- Simplification: \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \)
- Answer: \( 2 \)

3. Expression: \( \sqrt{9} \)
- Simplification: \( \sqrt{9} = 3 \)
- Answer: \( 3 \)

4. Expression: \( \sqrt[3]{27} \)
- Simplification: \( \sqrt[3]{27} = 3 \) because \( 3^3 = 27 \)
- Answer: \( 3 \)

5. Expression: \( \sqrt{16} \)
- Simplification: \( \sqrt{16} = 4 \)
- Answer: \( 4 \)

6. Expression: \( \sqrt[3]{64} \)
- Simplification: \( \sqrt[3]{64} = 4 \) because \( 4^3 = 64 \)
- Answer: \( 4 \)

7. Expression: \( \left( \sqrt{2} \right)^2 \)
- Simplification: \( \left( \sqrt{2} \right)^2 = 2 \)
- Answer: \( 2 \)

8. Expression: \( \sqrt{\frac{1}{4}} \)
- Simplification: \( \sqrt{\frac{1}{4}} = \frac{1}{2} \)
- Answer: \( \frac{1}{2} \)

9. Expression: \( \sqrt{0} \)
- Simplification: \( \sqrt{0} = 0 \)
- Answer: \( 0 \)

10. Expression: \( \sqrt[3]{-8} \)
- Simplification: \( \sqrt[3]{-8} = -2 \) because \( (-2)^3 = -8 \)
- Answer: \( -2 \)

11. Expression: \( \sqrt{-1} \)
- Simplification: \( \sqrt{-1} \) is not a real number; it is denoted as \( i \) in complex numbers.
- Answer: \( i \) (not a real number)

12. Expression: \( \sqrt[3]{\frac{1}{8}} \)
- Simplification: \( \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \) because \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
- Answer: \( \frac{1}{2} \)

13. Expression: \( \sqrt{4} \cdot \sqrt{9} \)
- Simplification: \( \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6 \)
- Answer: \( 6 \)

14. Expression: \( \sqrt{16} \cdot \sqrt{25} \)
- Simplification: \( \sqrt{16} \cdot \sqrt{25} = 4 \cdot 5 = 20 \)
- Answer: \( 20 \)

15. Expression: \( \sqrt{4} + \sqrt{9} \)
- Simplification: \( \sqrt{4} + \sqrt{9} = 2 + 3 = 5 \)
- Answer: \( 5 \)

16. Expression: \( \sqrt{16} + \sqrt{25} \)
- Simplification: \( \sqrt{16} + \sqrt{25} = 4 + 5 = 9 \)
- Answer: \( 9 \)

#### True or False Statements:

1. Statement: \( \sqrt{a^2} = a \)
- Analysis: This is false. The correct statement is \( \sqrt{a^2} = |a| \), which accounts for both positive and negative values of \( a \).
- Answer: False

2. Statement: \( \sqrt{a^2} = -a \)
- Analysis: This is false. The correct statement is \( \sqrt{a^2} = |a| \), which is non-negative. It equals \( -a \) only if \( a \leq 0 \).
- Answer: False

3. Statement: \( \sqrt{a^2} = |a| \)
- Analysis: This is true. By definition, the square root of a squared term is the absolute value of the original term.
- Answer: True

4. Statement: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)
- Analysis: This is true for non-negative \( a \) and \( b \). The property holds because \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
- Answer: True

5. Statement: \( \sqrt{a} + \sqrt{b} = \sqrt{a+b} \)
- Analysis: This is false. The square root of a sum is not equal to the sum of the square roots. For example, \( \sqrt{4} + \sqrt{9} = 2 + 3 = 5 \), but \( \sqrt{4+9} = \sqrt{13} \neq 5 \).
- Answer: False

6. Statement: \( \sqrt{a^2 + b^2} = a + b \)
- Analysis: This is false. The expression \( \sqrt{a^2 + b^2} \) does not simplify to \( a + b \). For example, if \( a = 3 \) and \( b = 4 \), then \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \), but \( 3 + 4 = 7 \).
- Answer: False

7. Statement: \( \sqrt{a^2} + \sqrt{b^2} = a + b \)
- Analysis: This is false. The left-hand side simplifies to \( |a| + |b| \), which is not necessarily equal to \( a + b \). For example, if \( a = -3 \) and \( b = 4 \), then \( \sqrt{(-3)^2} + \sqrt{4^2} = 3 + 4 = 7 \), but \( -3 + 4 = 1 \).
- Answer: False

8. Statement: \( \sqrt{a^2} + \sqrt{b^2} = |a| + |b| \)
- Analysis: This is true. By definition, \( \sqrt{a^2} = |a| \) and \( \sqrt{b^2} = |b| \), so their sum is \( |a| + |b| \).
- Answer: True

Final Answers:

1. \( 2 \)
2. \( 2 \)
3. \( 3 \)
4. \( 3 \)
5. \( 4 \)
6. \( 4 \)
7. \( 2 \)
8. \( \frac{1}{2} \)
9. \( 0 \)
10. \( -2 \)
11. \( i \)
12. \( \frac{1}{2} \)
13. \( 6 \)
14. \( 20 \)
15. \( 5 \)
16. \( 9 \)

True or False:
1. False
2. False
3. True
4. True
5. False
6. False
7. False
8. True

Final Boxed Answer:

\[
\boxed{
\begin{array}{ll}
1. & 2 \\
2. & 2 \\
3. & 3 \\
4. & 3 \\
5. & 4 \\
6. & 4 \\
7. & 2 \\
8. & \frac{1}{2} \\
9. & 0 \\
10. & -2 \\
11. & i \\
12. & \frac{1}{2} \\
13. & 6 \\
14. & 20 \\
15. & 5 \\
16. & 9 \\
\text{True or False:} & \\
1. & \text{False} \\
2. & \text{False} \\
3. & \text{True} \\
4. & \text{True} \\
5. & \text{False} \\
6. & \text{False} \\
7. & \text{False} \\
8. & \text{True} \\
\end{array}
}
\]