Laws of exponents — review with real exponents

3.2.2 · Maths › Exponentials & Logarithms

HUM actually define kya kar rahe hain?

Laws (aur WHY har ek forced hai)

Derivation 1 — Product law $a^m{a}^n=a^{m+n}$ (pehle integers ke liye)

Integers se kyun shuru karte hain? Kyunki wahan "copies" literally hota hai, toh hum ise prove kar sakte hain, phir force karte hain ki yeh reals ke liye bhi kaam kare.

$a^m{a}^n=\underset{m}{\underbrace{(a\cdots a)}}\underset{n}{\underbrace{(a\cdots a)}}=\underset{m+n}{\underbrace{a\cdots a}}=a^{m+n}.$ Yeh step kyun? Multiplication associative hoti hai — $m$ copies ka ek pile phir $n$ copies ka ek pile milke bas $m+n$ copies ka ek pile ban jaata hai. Koi naya rule nahi, bas dobara ginti.

Derivation 2 — $a^0=1$ kyun? (forced hai, choose nahi kiya)

Hum chahte hain ki product law hold kare. $a^m{a}^n=a^{m+n}$ mein $n=0$ rakhte hain: $a^m\cdot a^0=a^{m+0}=a^m\Rightarrow a^0=\frac{a^m}{a^m}=1.$ Yeh step kyun? Humne $a^0=1$ apni marzi se decide nahi kiya — yeh ek hi value hai jo product law ko consistent rakhti hai. Yeh Derivation-from-scratch hai: definition maangi jaati hai law ke dwara.

Derivation 3 — $a^{-n}=1/a^n$ kyun? (forced)

$m=n,n\to -n$ rakhte hain: hum chahte hain $a^n{a}^{-n}=a^{n+(-n)}=a^0=1$. Isliye $a^{-n}=\frac{1}{a^n}.$ Yeh step kyun? Negative exponents define kiye jaate hain woh value ke roop mein jo unhe multiplicative inverse banaye — taaki law survive kare.

Derivation 4 — $a^{1/q}=\sqrt[q]{a}$ kyun? (power law se forced)

Hum chahte hain $(a^{1/q}{)}^q=a^{(1/q)\cdot q}=a^1=a$. Toh $a^{1/q}$ woh number hai jiska $q$-th power $a$ ho — yaani $q$-th root. $a^{1/q}=\sqrt[q]{a}.$ Yeh step kyun? $(a^m{)}^n=a^{mn}$ fractional exponents ke liye tabhi kaam kar sakta hai jab fractional powers roots hon.

Derivation 5 — Irrational $x$ tak extend karna

$x$ irrational ho (maano $\sqrt{2}$), toh use rationals se squeeze karo: $1.4,1.41,1.414,\cdots \to \sqrt{2}$. Har $a^{1.41}$ etc. defined hai (rational). Kyunki $a>0$ ke liye $a^x$ continuous aur monotonic hai, yeh values ek single number par converge karti hain, jise hum $a^{\sqrt{2}}$ naam dete hain. Saare laws limit tak pass hote hain kyunki limits $+,\times$ ko respect karti hain.

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

• Rational Exponents and Radicals — $a^{p/q}=\sqrt[q]{a^p}$ ka bridge.
• Exponential Functions and their Graphs — jahan $a^x$ ki continuity rehti hai.
• Logarithms — Definition — logs exponents ko undo karte hain; laws mirror karte hain.
• Binomial Theorem — $(a+b{)}^n$ expand karne ka sahi tarika.
• Limits and Continuity — irrational exponents ko justify karta hai.
• Number e and Natural Exponential — special base jo in laws par bani hai.

Concept Map