Exponent Rules

Introduction

• Exponent rules govern expressions like 2^5 or X^n.
• 2^5 is shorthand for 2 * 2 * 2 * 2 * 2 (5 times).
• X^n is X multiplied by itself N times.
• Base: number being multiplied (bottom).
• Exponent/Powers: number of times the base is multiplied (top).

Product Rule

• X^n * X^m = X^(n+m).
• Example: 2^3 * 2^4 = 2^7.
• Simply add the exponents if the bases are the same.

Quotient Rule

• X^n / X^m = X^(n-m).
• Example: 3^6 / 3^2 = 3^4.
• Simply subtract the exponents if the bases are the same.

Power Rule

• (X^n)^m = X^(n*m).
• Example: 5^4^3 = 5^12.
• When raising a power to another power, multiply the exponents.

Zero Exponent Rule

• Anything to the 0th power equals 1.
• Example: 2^0 = 1.
• Based on the rule: X^n / X^n = X^(n-n)= X^0 = 1.

Negative Exponent Rule

• X^(-n) = 1 / X^n.
• Example: 5^(-7) = 1 / 5^7.
• Based on product rule: X^n * X^(-n) = X^0 = 1.

Fractional Exponent

• X^(1/n) = Nth root of X.
• Example: 64^(1/3) = cube root of 64 = 4.
• Example: 9^(1/2) = square root of 9 = 3.

Distributing Exponents Over Product

• (X * Y)^n = X^n * Y^n.
• Example: (5 * 7)^3 = 5^3 * 7^3.

Distributing Exponents Over Quotient

• (X / Y)^n = X^n / Y^n.
• Example: (2/7)^5 = 2^5 / 7^5.

Important Notes

• Exponents do not distribute over addition/subtraction.
• Example: (A + B)^n ≠ A^n + B^n.
• Example: 2+3)^2 ≠ 2^2 + 3^2.

Summary of Rules

1. Product Rule
2. Quotient Rule
3. Power Rule
4. Zero Exponent
5. Negative Exponent
6. Fractional Exponent
7. Distribution over Product
8. Distribution over Quotient