Laws of Exponents for Real Numbers

• Extend exponent laws from whole numbers to integer and rational exponents.
• Apply a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^m ÷ a^n = a^(m−n).
• Define a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m.
• Simplify expressions with fractional exponents and negative exponents.
• Solve NCERT-style simplification problems with a > 0.

1. Product law — a^m × a^n = a^(m+n) (same base a > 0).
2. Quotient law — a^m ÷ a^n = a^(m−n).
3. Power law — (a^m)^n = a^(mn).
4. Zero exponent — a^0 = 1 for a ≠ 0.
5. Negative exponent — a^(−n) = 1/a^n.
6. Fractional exponent — a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m).
7. Examples — 8^(1/3) = 2; 16^(3/4) = (2)^3 = 8; 27^(−2/3) = 1/9.
8. Product of powers — (ab)^n = a^n b^n.
9. NCERT restriction — For rational exponents in Class 9, base a is taken positive unless stated.
10. Applications — Scientific notation, compound interest formulas, square-root as power 1/2.

NCERT: Simplify (125)^(−1/3) × (625)^(1/4)

Step 1 — 125 = 5³ → 125^(−1/3) = 5^(3×(−1/3)) = 5^(−1) = 1/5
Step 2 — 625 = 5⁴ → 625^(1/4) = 5^(4×1/4) = 5^1 = 5
Step 3 — Product: (1/5) × 5 = 1
Answer: 1
Alternative: express both as powers of 5 first, then add exponents

• Applying laws when bases differ (2³ × 3² cannot combine exponents).
• Writing a^(1/2) as 1/a² (correct: √a, not 1/a²).
• Forgetting a^0 = 1 only for a ≠ 0.
• Taking even root of negative without context (⁴√(−16) not real).
• (a+b)^n ≠ a^n + b^n — cannot distribute exponent over addition.

• Simplify: 2^(−3) × 2^5.
• Evaluate 81^(1/2).
• Write √5 as a power of 5.
• Simplify (3^2)^4.
• Find 16^(3/4).

Open the Practice tab for graded questions on Laws of Exponents for Real Numbers.