#A03 · Algebra & Number Theory · Lv.1View in 3D Mind-Map

Laws of Exponents & Radicals

Nicole Oresme & René Descartes · 1637 CE

a^m \cdot a^n = a^{m+n},\quad a^{m/n} = \sqrt[n]{a^m}

(a^m)^n = a^{mn},\quad a^{-n} = \frac{1}{a^n}

Unifies repeated multiplication and n-th roots into rational and real exponents, forming the basis for exponential functions, logarithms, and power-rule derivatives.

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Base a

Power m

Root n

8^{2/3} = \sqrt[3]{8^{2}} = 4.0000

a^(m/n)

4.0000

유리수 지수 m/n

0.667

역수 a^(-m/n)

0.2500

Prerequisite Path (2 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

🚀 Next Unlocks (Derived Formulas) (7)

#A04Quadratic Formula & Discriminant #A06Logarithm Laws & Change of Base #A07Arithmetic & Geometric Series Sum #A08Binomial Theorem & Pascal's Triangle #A10Modular Arithmetic & Fermat's Little Theorem #G01Pythagorean Theorem #C01Limits, Continuity & Euler's Number e

Geometric & Intuitive Insight

거듭제곱근 √a 는 a^(1/2)과 같아서, 두 번 곱하면 a^(1/2 + 1/2) = a^1 = a가 됩니다. 루트 기호를 분수 지수로 바꾸면 모든 연산이 지수의 사칙연산으로 단순화됩니다.

Step-by-Step Derivation

Step 1.1단계: m번 곱한 a^m 과 n번 곱한 a^n 을 곱하면 총 (m+n)번 곱해지므로 a^(m+n)
Step 2.2단계: (a^(1/n))^n = a^(n/n) = a^1 이므로 a^(1/n) 은 n제곱하여 a가 되는 수 ⁿ√a
Step 3.3단계: a^0 = a^(n-n) = a^n / a^n = 1 (단, a ≠ 0)

AI & Coding Real-World Use

IEEE 754 floating-point exponent encoding and 3D shader gamma correction (c^(1/2.2))

#IEEE 754 Float#Shader Gamma Correction#Exponential Scaling

const gammaCorrect = (c: number, g = 2.2) => Math.pow(c, 1 / g);

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