Question

apply all exponent rules quick check

Answer 1

Certainly! Here’s a quick overview of the exponent rules along with some examples for each:

### Exponent Rules

1. **Product of Powers Rule**:
\[
a^m \times a^n = a^{m+n}
\]
- Example: \( x^3 \times x^4 = x^{3+4} = x^7 \)

2. **Quotient of Powers Rule**:
\[
\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)
\]
- Example: \( \frac{y^5}{y^2} = y^{5-2} = y^3 \)

3. **Power of a Power Rule**:
\[
(a^m)^n = a^{m \cdot n}
\]
- Example: \( (z^2)^3 = z^{2 \cdot 3} = z^6 \)

4. **Power of a Product Rule**:
\[
(ab)^n = a^n \times b^n
\]
- Example: \( (2x)^3 = 2^3 \times x^3 = 8x^3 \)

5. **Power of a Quotient Rule**:
\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)
\]
- Example: \( \left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16} \)

6. **Zero Exponent Rule**:
\[
a^0 = 1 \quad (a \neq 0)
\]
- Example: \( 5^0 = 1 \)

7. **Negative Exponent Rule**:
\[
a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
\]
- Example: \( x^{-3} = \frac{1}{x^3} \)

### Quick Check Examples

1. **Calculate** \( 3^2 \times 3^4 \):
\[
= 3^{2+4} = 3^6 = 729
\]

2. **Calculate** \( \frac{a^6}{a^2} \):
\[
= a^{6-2} = a^4
\]

3. **Calculate** \( (2^3)^2 \):
\[
= 2^{3 \cdot 2} = 2^6 = 64
\]

4. **Calculate** \( (xy)^3 \):
\[
= x^3 y^3
\]

5. **Calculate** \( \left(\frac{4}{5}\right)^3 \):
\[
= \frac{4^3}{5^3} = \frac{64}{125}
\]

6. **Calculate** \( 7^0 \):
\[
= 1
\]

7. **Calculate** \( 2^{-3} \):
\[
= \frac{1}{2^3} = \frac{1}{8}
\]

Feel free to ask if you need further examples or clarifications on any specific exponent rules!