Chapter 1: Number Systems

Laws of Exponents for Real Numbers

The Four Fundamental Laws

For a>0a > 0 and rational exponents p,qp, q:

Law 1: Product of Same Base

ap×aq=ap+qa^p \times a^q = a^{p+q}

Example: 23×24=23+4=27=1282^3 \times 2^4 = 2^{3+4} = 2^7 = 128

With fractions: 51/2×51/3=51/2+1/3=55/65^{1/2} \times 5^{1/3} = 5^{1/2 + 1/3} = 5^{5/6}

Law 2: Power of a Power

(ap)q=apq(a^p)^q = a^{pq}

Example: (32)4=32×4=38(3^2)^4 = 3^{2 \times 4} = 3^8

With fractions: (41/2)3=43/2=(4)3=8(4^{1/2})^3 = 4^{3/2} = (\sqrt{4})^3 = 8

Law 3: Quotient of Same Base

apaq=ap−q\frac{a^p}{a^q} = a^{p-q}

Example: 5753=57−3=54=625\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625

With fractions: 71/271/4=71/2−1/4=71/4\frac{7^{1/2}}{7^{1/4}} = 7^{1/2 - 1/4} = 7^{1/4}

Law 4: Product of Same Power

ap×bp=(ab)pa^p \times b^p = (ab)^p

Example: 32×42=(3×4)2=122=1443^2 \times 4^2 = (3 \times 4)^2 = 12^2 = 144

With fractions: 91/2×41/2=(36)1/2=69^{1/2} \times 4^{1/2} = (36)^{1/2} = 6

Special Values

ExpressionValueWhy
a0a^01Definition
a−na^{-n}1/an1/a^nNegative exponent = reciprocal
a1/na^{1/n}an\sqrt[n]{a}Fractional exponent = root
am/na^{m/n}(an)m(\sqrt[n]{a})^mCombine root and power

Common Mistakes to Avoid

❌ Wrong✅ Correct
23×33=632^3 \times 3^3 = 6^3 ❌(2×3)3=63(2 \times 3)^3 = 6^3 ✓ Only when bases same
(23)4=27(2^3)^4 = 2^7 ❌(23)4=212(2^3)^4 = 2^{12} ✓ Multiply exponents
23+24=272^3 + 2^4 = 2^7 ❌23+24=8+16=242^3 + 2^4 = 8 + 16 = 24 ✓ Cant combine addition