M2 - Surds and Indices

OCR MEI AS Mathematics – Chapter 2: Surds and Indices

1. Indices (Powers/Exponents)

• An index is the power to which a number is raised.

  • Example: $a^n$ means "a multiplied by itself n times".

1.1 Basic Laws of Indices

• For any real numbers $a <br>eq 0$ and integers $m, n$:

  • Multiply same base:

  • Rule: $a^m * a^n = a^{(m+n)}$

  • Example: $x^2 * x^3 = x^5$

  • Divide same base:

  • Rule: $rac{a^m}{a^n} = a^{(m-n)}$

  • Example: $rac{x^5}{x^2} = x^3$

  • Power of a power:

  • Rule: $(a^m)^n = a^{(m*n)}$

  • Example: $(x^2)^3 = x^6$

  • Power of a product:

  • Rule: $(ab)^n = a^n * b^n$

  • Example: $(2x)^3 = 8x^3$

  • Power of a quotient:

  • Rule: $rac{(a/b)^n} = rac{a^n}{b^n}$

  • Example: $(x/2)^2 = rac{x^2}{4}$

  • Zero index:

  • Rule: $a^0 = 1$

  • Example: $5^0 = 1$

  • Negative index:

  • Rule: $a^{-n} = rac{1}{a^n}$

  • Example: $2^{-3} = rac{1}{8}$

  • Fractional index:

  • Rule: $a^{(m/n)} = ext{nth<em>root}(a^m) = ( ext{nth</em>root}(a))^m$

  • Example: $8^{(2/3)} = ( ext{cube_root}(8))^2 = 2^2 = 4$

2. Surds

• A surd is a root that cannot be simplified to remove the root.

  • Examples: $ext{√2}, ext{√3}, ext{√5}$ are surds.

  • Non-example: $ext{√4} = 2$ is not a surd.

2.1 Simplifying Surds

1. Product rule:

   • $ext{√a} * ext{√b} = ext{√(a*b)}$

2. Quotient rule:

   • $rac{ ext{√a}}{ ext{√b}} = ext{√(a/b)}$

3. Simplifying square roots by factorizing:

   • $ext{√(a*b^2)} = b ext{√a}$

4. Rationalising the denominator:

   • a) Single term:

     • $rac{1}{ ext{√a}} = rac{ ext{√a}}{a}$

   • b) Two terms:

     • $rac{1}{(a+ ext{√b})} = rac{(a- ext{√b})}{(a^2 - b)}$

3. Operations with Surds

1. Addition/Subtraction:

   • Only like surds can be added or subtracted.

   • Example: $3 ext{√2} + 5 ext{√2} = 8 ext{√2}$

2. Multiplication:

   • Use product rule.

   • Example: $ext{√2} * 3 ext{√2} = 6$

3. Division:

   • Use rationalisation.

   • Example: $rac{5}{ ext{√3}} = rac{5 ext{√3}}{3}$

4. Combining Surds and Indices:

   • Convert surds to fractional indices:

     • $ext{√a} = a^{(1/2)}$

     • $a = a^{(1/3)}$

   • Examples:

     • $( ext{√a})^4 = (a^{(1/2)})^4 = a^2$

     • $ext{√a} * a = a^{(1/2)} * a^{(1/3)} = a^{(5/6)}$

4. Logarithms (Optional)

• Definition:

  • $ext{log}_a x = n$ means $a^n = x$

• Laws:

  • $ext{log}<em>a(x*y) = ext{log}</em>a x + ext{log}_a y$

  • $ext{log}<em>a(x/y) = ext{log}</em>a x - ext{log}_a y$

  • $ext{log}<em>a(x^n) = n * ext{log}</em>a x$

• Change of base:

  • $ext{log}<em>a x = rac{ ext{log}</em>b x}{ ext{log}_b a}$

• Example:

  • $ext{log}_{10} 100 = 2$ because $10^2 = 100$

5. Key Exam Tips

1. Simplify surds fully.

2. Convert roots to fractional indices in algebra.

3. Rationalise denominators.

4. Check your powers and indices carefully.

5. Practice combining surds and indices in one step.

6. Summary of Equations/Rules

• Indices:

  • $a^m * a^n = a^{(m+n)}$

  • $rac{a^m}{a^n} = a^{(m-n)}$

  • $(a^m)^n = a^{(mn)}$

  • $(ab)^n = a^n * b^n$

  • $rac{(a/b)^n}{b^n} = rac{a^n}{b^n}$

  • $a^0 = 1$

  • $a^{-n} = rac{1}{a^n}$

  • $a^{(m/n)} = ext{nth<em>root}(a^m) = ( ext{nth</em>root}(a))^m$

• Surds:

  • $ext{√a} * ext{√b} = ext{√(a*b)}$

  • $rac{ ext{√a}}{ ext{√b}} = ext{√(a/b)}$

  • $ext{√(a*b^2)} = b ext{√a}$

  • $rac{1}{ ext{√a}} = rac{ ext{√a}}{a}$

  • $rac{1}{(a+ ext{√b})} = rac{(a- ext{√b})}{(a^2-b)}$

• Combine Surds & Indices:

  • $ext{√a} = a^{(1/2)}$, $a = a^{(1/3)}$

  • $( ext{√a})^4 = a^2$

  • $ext{√a} * a = a^{(5/6)}$