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Exponent Rules



Basic Power Rules
Special Exponent Rules
Advanced Power Rules
Exponential Function Rules
Root and Radical Rules
Logarithmic Power Relations






Basic Power Rules

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Product Rule
am×an=am+na^m \times a^n = a^{m+n}
When multiplying powers with the same base, add the exponents
Quotient Rule
aman=am−n\frac{a^m}{a^n} = a^{m-n}
When dividing powers with the same base, subtract the exponents
Power of a Power Rule
(am)n=am⋅n\left(a^m\right)^n = a^{m \cdot n}
When raising a power to another power, multiply the exponents
Power of a Product Rule
(ab)n=an⋅bn\left(ab\right)^n = a^n \cdot b^n
When raising a product to a power, raise each factor to that power
Power of a Quotient Rule
(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
When raising a quotient to a power, raise both numerator and denominator to that power

Special Exponent Rules

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Zero Exponent Rule
a0=1a^0 = 1
Any non-zero number raised to the power of zero equals 1 (where a≠0a \neq 0)
Negative Exponent Rule
a−n=1ana^{-n} = \frac{1}{a^n}
A negative exponent means the reciprocal of the positive exponent
Fractional Exponent Rule
amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m
A fractional exponent represents roots and powers combined
Unit Base Rule
1n=11^n = 1
One raised to any power equals one
Unit Exponent Rule
a1=aa^1 = a
Any number raised to the first power equals itself

Advanced Power Rules

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Distributive Property with Powers
an⋅bn=(ab)na^n \cdot b^n = (ab)^n
Product of powers with same exponent equals power of the product
Quotient of Powers with Same Exponent
anbn=(ab)n\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n
Quotient of powers with same exponent equals power of the quotient
Power of a Negative Base (Even Exponent)
(−a)2n=a2n(-a)^{2n} = a^{2n}
Negative base raised to even power gives positive result
Power of a Negative Base (Odd Exponent)
(−a)2n+1=−a2n+1(-a)^{2n+1} = -a^{2n+1}
Negative base raised to odd power gives negative result

Exponential Function Rules

lawformulaexplanation
Exponential Product Rule
bx⋅by=bx+yb^x \cdot b^y = b^{x+y}
Product of exponentials with same base adds exponents
Exponential Quotient Rule
bxby=bx−y\frac{b^x}{b^y} = b^{x-y}
Quotient of exponentials with same base subtracts exponents
Power of Exponential Rule
(bx)y=bxy\left(b^x\right)^y = b^{xy}
Power of an exponential multiplies the exponents
Exponential with Different Bases
ax⋅bx=(ab)xa^x \cdot b^x = (ab)^x
Product of exponentials with same exponent but different bases

Root and Radical Rules

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nth Root Definition
an=a1/n\sqrt[n]{a} = a^{1/n}
nth root expressed as fractional exponent
Product of Roots
an⋅bn=abn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}
Product of nth roots equals nth root of the product
Quotient of Roots
anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}
Quotient of nth roots equals nth root of the quotient
Power of a Root
(an)m=amn=am/n\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} = a^{m/n}
Power of an nth root can be expressed as fractional exponent
Root of a Root
anm=amn=a1/(mn)\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = a^{1/(mn)}
Nested roots multiply the indices

Logarithmic Power Relations

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Logarithm of a Power
log⁡b(an)=n⋅log⁡b(a)\log_b(a^n) = n \cdot \log_b(a)
Logarithm of a power brings the exponent as a coefficient
Power as Exponential
an=en⋅ln⁡(a)a^n = e^{n \cdot \ln(a)}
Any power can be expressed using natural exponential and logarithm
Change of Base for Powers
ax=bx⋅log⁡b(a)a^x = b^{x \cdot \log_b(a)}
Express power with different base using logarithms