Indices Rules
Indices (Exponents) Rules
Exponent of Zero
Rule: \( a^0 = 1 \) (for any \( a \neq 0 \)).
A non-zero base raised to the power of zero is always 1.
Example: \( 5^0 = 1 \).
Exponent of One
Rule: \( a^1 = a \).
A base raised to the power of one is itself.
Example: \( 7^1 = 7 \).
Product of Powers
Rule: \( a^m \times a^n = a^{m + n} \).
When multiplying exponential terms with the same base, add the exponents.
Example: \( x^2 \times x^3 = x^{2+3} = x^5 \).
Quotient of Powers
Rule: \(\frac{a^m}{a^n} = a^{m - n}\) (for \( a \neq 0 \)).
When dividing exponential terms with the same base, subtract the exponents.
Example: \(\frac{y^5}{y^2} = y^{5-2} = y^3\).
Power of a Power
Rule: \((a^m)^n = a^{mn}\).
When raising a power to another power, multiply the exponents.
Example: \(\bigl(x^2\bigr)^3 = x^{2 \times 3} = x^6\).
Negative Exponent
Rule: \( a^{-n} = \frac{1}{a^n} \) (for \( a \neq 0 \)).
A negative exponent indicates the reciprocal of the base raised to the corresponding positive power.
Example: \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).
