Mathematics Grade 9 - Notes from Page 1-11

Mathematics Grade 9

  • Indices
    • Deals with the powers and roots of numbers and variables.
    • Related to ICT: Megabytes, Gigabytes, and Terabytes (Mega = 10610^6, Giga = 10910^9, Tera = 101210^{12}).
    • Revisiting Laws of Indices before solving equations.

Historical Fact

  • Nicomachus of Gerasa's number pattern:
    • 1=1=131 = 1 = 1^3
    • 3+5=8=233 + 5 = 8 = 2^3
    • 7+9+11=27=337 + 9 + 11 = 27 = 3^3
    • 13+15+17+19=64=4313 + 15 + 17 + 19 = 64 = 4^3

Multiplication Law

  • am×an=a(m+n)a^m \times a^n = a^{(m+n)}
    • Example:
      • 24×22=24+2=262^4 \times 2^2 = 2^{4+2} = 2^6
      • x4×y2×x3=x4+3y2=x7y2x^4 \times y^2 \times x^3 = x^{4+3} y^2 = x^7 y^2
      • 5x4×3x2=(5×3)x4+2=15x65x^4 \times 3x^2 = (5 \times 3)x^{4+2} = 15x^6

Laws of Indices

  • Caution:
    • 23×24≠43+42^3 \times 2^4 \neq 4^{3+4}
    • 23×24=23+4=272^3 \times 2^4 = 2^{3+4} = 2^7
    • ax+ay≠ax+ya^x + a^y \neq a^{x+y}
    • 32+33≠32+33^2 + 3^3 \neq 3^{2+3}
    • 32+33=9+273^2 + 3^3 = 9 + 27

Division Law

  • am÷an=a(m−n),a≠0a^m \div a^n = a^{(m-n)}, a \neq 0
    • Example:
      • 27÷23=27−3=242^7 \div 2^3 = 2^{7-3} = 2^4
      • x4y5÷x3y2=x4−3y5−2=xy3x^4 y^5 \div x^3 y^2 = x^{4-3} y^{5-2} = xy^3
      • 18x53x2=6x5−2=6x3\frac{18x^5}{3x^2} = 6x^{5-2} = 6x^3

Power Law

  • (am)n=amn(a^m)^n = a^{mn}
    • Example:
      • (32)4=32×4=38(3^2)^4 = 3^{2\times4} = 3^8
      • (m3)5=m3×5=m15(m^3)^5 = m^{3\times5} = m^{15}
      • 3(p2)4=3p2×4=3p83(p^2)^4 = 3p^{2\times4} = 3p^8

Zero Index

  • a0=1,a≠0a^0 = 1, a \neq 0
    • Example:
      • 30=13^0 = 1
      • 2m0=2(1)=22m^0 = 2(1) = 2
      • (2m)0=1(2m)^0 = 1

Recall

  • Using division law, 3232=30=1\frac{3^2}{3^2} = 3^0 = 1

Rules

  • (Rule 1) (ab)n=anbn(ab)^n = a^n b^n
    * (ab)3=(ab)×(ab)×(ab)=a×b×a×b×a×b=a3b3(ab)^3 = (ab) \times (ab) \times (ab) = a \times b \times a \times b \times a \times b = a^3 b^3
  • (Rule 2) (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}
    * (ab)2=ab×ab=a×ab×b=a2b2(\frac{a}{b})^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a \times a}{b \times b} = \frac{a^2}{b^2}

Example

  • Simplify the following
    • (2p)3=23p3=8p3(2p)^3 = 2^3 p^3 = 8p^3
    • (1x)2=12x2=1x2(\frac{1}{x})^2 = \frac{1^2}{x^2} = \frac{1}{x^2}
    • (x3y2)5=(x3)5(y2)5=x15y10(x^3 y^2)^5 = (x^3)^5 (y^2)^5 = x^{15} y^{10}
    • (x2)4(y3)4=x8y12\frac{(x^2)^4}{(y^3)^4} = \frac{x^8}{y^{12}}

Negative Indices

  • a−n=1an,a≠0a^{-n} = \frac{1}{a^n}, a \neq 0
    • Example:
      • 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}
      • 4y−3=4y34y^{-3} = \frac{4}{y^3}
      • x−3×x−2=x−3+(−2)=x−5=1x5x^{-3} \times x^{-2} = x^{-3 + (-2)} = x^{-5} = \frac{1}{x^5}
      • 22÷27=22−7=2−5=125=1322^2 \div 2^7 = 2^{2-7} = 2^{-5} = \frac{1}{2^5} = \frac{1}{32}

Fractional Indices

  • a=a12\sqrt{a} = a^{\frac{1}{2}}
  • a3=a13\sqrt[3]{a} = a^{\frac{1}{3}}
  • In general, an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}
  • Note: amn=amn=(am)1n=amn\sqrt[n]{a^m} = a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}
  • Example:
    • 81=(81)12=34=32=9\sqrt{81} = (81)^{\frac{1}{2}} = \sqrt{3^4} = 3^2 = 9
    • 643=(26)13=22=4\sqrt[3]{64} = (2^6)^{\frac{1}{3}} = 2^2 = 4
    • (32)15=(25)15=21=2(32)^{\frac{1}{5}} = (2^5)^{\frac{1}{5}} = 2^1 = 2
    • (46)12=43=64(4^6)^{\frac{1}{2}} = 4^3 = 64

Equations Involving Indices

  • If ax=aya^x = a^y then x=yx = y
  • Solve the following equations.
    • 5x=535^x = 5^3
      • Since the bases are the same, x=3x=3.
    • 32x+1=323^{2x+1} = 3^2
      • Since the bases are the same, 2x+1=22x+1=2, 2x=12x = 1, x=12x = \frac{1}{2}.
    • m4=34m^4 = 3^4
      • Comparing indices and bases we get, m=3m=3.

Note

  • 2x=124=2−42x = \frac{1}{2^4} = 2^{-4}, x=−4x = -4

Summary of Indices

  1. Multiplication law: am×an=a(m+n)a^m \times a^n = a^{(m+n)}
  2. Division law: am÷an=a(m−n)a^m \div a^n = a^{(m-n)}
  3. Power law: (am)n=amn(a^m)^n = a^{mn}
  4. Zero index: a0=1a^0 = 1
  5. Negative index: a−n=1ana^{-n} = \frac{1}{a^n}
  6. Rule 1: (ab)n=anbn(ab)^n = a^n b^n
  7. Rule 2: (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}
  8. Fractional indices:
    • Square roots: a=a12\sqrt{a} = a^{\frac{1}{2}}
    • Cube roots: a3=a13\sqrt[3]{a} = a^{\frac{1}{3}}
  9. Equations involving indices:
    • If ax=aya^x = a^y then x=yx=y
    • Or If xm=ymx^m = y^m then x=yx = y