Exponent Rules & Simplifying Expressions

Exponent Rules

• Product Rule: When multiplying two or more exponential expressions with the same base, add the exponents.
  • Example: a^m × a^n = a^(m+n)
• Quotient Rule: When dividing two exponential expressions with the same base, subtract the exponents.
  • Example: a^m ÷ a^n = a^(m-n)
• Power Rule: When raising an exponential expression to a power, multiply the exponents.
  • Example: (a^m)^n = a^(mn)

Simplifying Expressions

• Simplify expressions by combining like terms and applying the exponent rules.
• Example: 2^3 × 2^5 = 2^(3+5) = 2^8
• Example: 3^2 × 3^4 ÷ 3^2 = 3^(2+4-2) = 3^4

Index Laws

• Zero Index: Any number raised to the power of 0 is 1.
  • Example: a^0 = 1
• Negative Index: A negative index is equivalent to a reciprocal with a positive index.
  • Example: a^(-n) = 1/a^n
• Fractional Index: A fractional index is equivalent to a root of the base.
  • Example: a^(1/n) = nth root of a

Squaring And Cubing

• Squaring: When raising a number to the power of 2, multiply it by itself.
  • Example: a^2 = a × a
• Cubing: When raising a number to the power of 3, multiply it by itself twice.
  • Example: a^3 = a × a × a
• Simplifying Squares and Cubes: Use the exponent rules to simplify expressions involving squares and cubes.
  • Example: 2^3 × 2^2 = 2^(3+2) = 2^5
  • Example: (3^2)^3 = 3^(2×3) = 3^6

Exponent Rules

• Exponent rules are used to simplify exponential expressions
• Product Rule: Add exponents when multiplying exponential expressions with the same base
• Quotient Rule: Subtract exponents when dividing exponential expressions with the same base
• Power Rule: Multiply exponents when raising an exponential expression to a power

Simplifying Expressions

• Combine like terms and apply exponent rules to simplify expressions
• Simplify expressions by adding or subtracting exponents when the bases are the same
• Example: Combine 2^3 and 2^5 by adding exponents: 2^(3+5) = 2^8

Index Laws

• Zero Index Law: Any number raised to the power of 0 is 1
• Negative Index Law: A negative index is equivalent to a reciprocal with a positive index
• Fractional Index Law: A fractional index is equivalent to a root of the base

Squares and Cubes

• Squaring: Raising a number to the power of 2 is equivalent to multiplying it by itself
• Cubing: Raising a number to the power of 3 is equivalent to multiplying it by itself twice
• Simplify square and cube expressions using exponent rules
• Example: (3^2)^3 can be simplified using the power rule: 3^(2×3) = 3^6