Algebraic Expressions and Identities

• Indices, also known as exponents or powers, denote the number of times a base number is multiplied by itself
  • It is written as a base number with a superscript indicating the index.
• Algebraic techniques involve manipulating expressions using the rules of algebra to simplify or solve equations.

Simplifying Algebraic Expressions

• Combining Like Terms: Terms with the same variable and exponent can be combined by adding or subtracting their coefficients.
  • Example: (3x + 5x = 8x)
• Distributive Property: Distribute a term across multiple terms within parentheses.
  • Example: (a(b + c) = ab + ac)
• Factoring: Expressing an expression as a product of its factors.
  • Example: (x^2 + 5x + 6 = (x + 2)(x + 3))

Algebraic Identities

• Algebraic identities are equations that are always true regardless of the values of the variables involved.
• Important algebraic identities include:
  • ((a + b)^2 = a^2 + 2ab + b^2)
  • ((a - b)^2 = a^2 - 2ab + b^2)
  • ((a + b)(a - b) = a^2 - b^2)
  • ((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3)
  • ((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3)
  • (a^3 + b^3 = (a + b)(a^2 - ab + b^2))
  • (a^3 - b^3 = (a - b)(a^2 + ab + b^2))

Exponential Equations

• Exponential equations are equations in which the variable occurs in the exponent.
• To solve exponential equations:
  • If possible, express both sides of the equation with the same base:
    • If (a^x = a^y), then (x = y)
  • Use logarithms to solve for the variable if expressing both sides with the same base is not feasible.
    • Example: (2^x = 7) can be solved by taking the logarithm of both sides.

Properties of Indices

• Product of Powers: When multiplying like bases, add the exponents.
  • (a^m \cdot a^n = a^{m+n})
• Quotient of Powers: When dividing like bases, subtract the exponents.
  • (\frac{a^m}{a^n} = a^{m-n})
• Power of a Power: When raising a power to a power, multiply the exponents.
  • ((a^m)^n = a^{mn})
• Power of a Product: The power of a product is the product of the powers.
  • ((ab)^n = a^n b^n)
• Power of a Quotient: The power of a quotient is the quotient of the powers.
  • ((\frac{a}{b})^n = \frac{a^n}{b^n})
• Zero Exponent: Any non-zero number raised to the power of 0 is 1.
  • (a^0 = 1), for (a \neq 0)
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent.
  • (a^{-n} = \frac{1}{a^n})
• Fractional Exponent: A fractional exponent indicates a root.
  • (a^{\frac{1}{n}} = \sqrt[n]{a})
  • (a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m)