Simplifying Exponents

Basic Properties of Exponents

Multiplying Common Bases: When multiplying with a common base, add the exponents.
- Example: (x^4 \times x^5 = x^{4+5} = x^9)
Dividing Common Bases: When dividing with a common base, subtract the exponents.
- Example: (\frac{x^7}{x^3} = x^{7-3} = x^4)
Raising an Exponent: When raising one exponent to another, multiply the exponents.
- Example: ((x^3)^4 = x^{3 \times 4} = x^{12})
Zero Exponent: Anything raised to the zero power equals 1.
- Example: (x^0 = 1)
Negative Exponents: Move to the opposite side of the fraction and change the sign of the exponent.
- Example: (x^{-3} = \frac{1}{x^3})

Calculate (-32):
- (-32 = -3 \times -3 = -9)
- (-32 = \text{negative 1} \times 3^2 = -9)

First Example: (x^3 \text{ raised to the 5th power} ) and (x^7 / x^{12})
- Solution:
  - For multiplication: ((x^3)^5 = x^{15})
  - For division: (x^{7-12} = x^{-5} = \frac{1}{x^5})
Second Example: (3x^2) raised to the 3rd power:
- ((3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6)
Third Example: (-2x^3y^4) raised to the second power:
- ((-2x^3y^4)^2 = (-2)^2 \times (x^3)^2 \times (y^4)^2 = 4x^6y^8)
Multiplying Expressions: (5x^3 \times 4x^7)
- Multiply coefficients: (5 \times 4 = 20)
- Add exponents of x: (x^{3+7} = x^{10})
- Result: (20x^{10})
Another Example: (24x^7y^3 / 8x^4y^{-2})
- Divide coefficients: (\frac{24}{8} = 3)
- Subtract exponents: (x^{7-4} = x^{3}) and (y^{3 - (-2)} = y^{5})
- Result: (3x^3y^5)
Final Example: (35x^3y^5 / 63x^4y^7)**
- Simplify coefficients: (\frac{35}{63} = \frac{5}{9})
- Subtract exponents: (x^{3-4} = x^{-1} = \frac{1}{x}) and (y^{5-7} = y^{-2} = \frac{1}{y^2})
- Result: (\frac{5}{9} \frac{1}{xy^2} = \frac{5}{9xy^2})
- Final answer: (\frac{5}{9y^2} \times x^{2 \text{ raised to the 2 power}} = \frac{5x^{2y^4}}{9})

When simplifying exponents, remember the rules of addition, subtraction, zero, and negative exponents.
Practice with multiple examples to gain a solid understanding.