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!