Question

variable expressions with exponents quick check

Answer 1

Sure! Here are some concepts and a quick check related to variable expressions with exponents:

### Key Concepts

1. **Exponent Rules**:
- **Product of Powers**: \( a^m \times a^n = a^{m+n} \)
- **Quotient of Powers**: \( \frac{a^m}{a^n} = a^{m-n} \) (where \( a \neq 0 \))
- **Power of a Power**: \( (a^m)^n = a^{m \cdot n} \)
- **Power of a Product**: \( (ab)^n = a^n b^n \)
- **Power of a Quotient**: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \) (where \( b \neq 0 \))

2. **Zero Exponent**:
- \( a^0 = 1 \) (where \( a \neq 0 \))

3. **Negative Exponent**:
- \( a^{-n} = \frac{1}{a^n} \) (where \( a \neq 0 \))

### Quick Check Examples

Let's evaluate and simplify the following expressions:

1. **Expression**: \( x^3 \times x^2 \)
**Simplified**:
\( x^{3+2} = x^5 \)

2. **Expression**: \( \frac{y^4}{y^2} \)
**Simplified**:
\( y^{4-2} = y^2 \)

3. **Expression**: \( (2x^2)^3 \)
**Simplified**:
\( 2^3 (x^2)^3 = 8x^{2 \cdot 3} = 8x^6 \)

4. **Expression**: \( (ab)^2 \)
**Simplified**:
\( a^2 b^2 \)

5. **Expression**: \( \frac{3x^{-1}}{y^2} \)
**Simplified**:
\( \frac{3}{xy^2} \) (since \( x^{-1} = \frac{1}{x} \))

6. **Expression**: \( (x^3y^2)^2 \)
**Simplified**:
\( x^{3\cdot2} y^{2\cdot2} = x^6 y^4 \)

### Practice Problems

Try simplifying the following:

1. \( a^2 \times a^5 \)
2. \( \frac{m^5}{m^3} \)
3. \( (3x^2y)^2 \)
4. \( (2^3 \cdot x^{-5})^2 \)
5. \( \left(\frac{p^4}{q^2}\right)^{-1} \)

Feel free to check your answers, and let me know if you need help!