Exponents

Product Rule

The product rule for exponents states that when multiplying two expressions with the same base, you can add the exponents:

  1. \( a^m \times a^n = a^{m+n} \)
  2. \( 2^3 \times 2^4 = 2^{3+4} = 2^7 = 128 \)
  3. \( (2x^2y)(-3x^3y^4) = -6x^5y^5 \)

Power Rule

The power rule for exponents states that when raising a power to another power, you multiply the exponents:

  1. \( (a^m)^n = a^{m \times n} \)
  2. \( (x^2)^3 = x^6 \)
  3. \( ((2y)^3)^2 = (2y)^6 \)

Quotient Rule

The quotient rule for exponents states that when dividing two expressions with the same base, you subtract the exponents:

  1. \( \frac{a^m}{a^n} = a^{m-n} \)
  2. \( \frac{x^5}{x^2} = x^3 \)
  3. \( \frac{y^7}{y^3} = y^4 \)
  4. \( \frac{10^6}{10^2} = 10^4 \)
  5. \( \frac{x^4y^6}{x^2y^3} = x^2y^3 \)
  6. \( \frac{10^9}{10^4} = 10^5 \)