Laws of integral indices

Skills and methods in this topic, easy to hard

1. 1Multiplying and dividing powersSame base: multiplying adds the indices, dividing subtracts them

   Methodx⁴ × x⁵ = x⁹; x⁷ ÷ x³ = x⁴

   Another way: Write them all outx³ × x² = (x × x × x) × (x × x) = x⁵

2. 2Powers of powers, products and quotients(a^m)ⁿ multiplies the indices; every factor in the bracket is raised

   Method(x³)⁴ = x¹²; (2x³)² = 4x⁶

   Another way: Write it as a product(x³)⁴ = x³ × x³ × x³ × x³ = x¹²

3. 3Zero and negative indicesa⁰ = 1, a⁻ⁿ = 1aⁿ; a negative index isn't a negative number

   Method2⁻³ = 1 ÷ 2³ = 18

   Another way: Follow the pattern down2¹ = 2 → 2⁰ = 1 → 2⁻¹ = 12 → 2⁻² = 14 → 2⁻³ = 18

4. 4Working with integral indicesWork the indices as directed numbers, then make them positive

   Methodx⁻³ × x⁵ = x²; (x⁻²)³ = x⁻⁶ = 1x⁶

   Another way: Move it across the bar firstx² ÷ x⁻⁴ = x² × x⁴ = x⁶

5. 5Writing powers with the same baseOne base first, then the laws; same base, equal indices

   Method4³ × 8² = 2⁶ × 2⁶ = 2¹²

   Another way: Work it out to check4³ × 8² = 64 × 64 = 4096 = 2¹²

6. 6Scientific notationa × 10ⁿ with 1 ≤ a < 10; the point moved left, index positive; right, negative

   Method3,400,000 = 3.4 × 10⁶; 0.00052 = 5.2 × 10⁻⁴

   Another way: Front numbers and powers apart(3 × 10⁴) × (5 × 10³) = 15 × 10⁷ = 1.5 × 10⁸

7. 7Binary numbersPlace values 1, 2, 4, 8, 16, …; expand to get the decimal

   Method1011₂ = 8 + 0 + 2 + 1 = 11

   Another way: Divide by 2 again and again13 ÷ 2 = 6 r 1, 6 ÷ 2 = 3 r 0, 3 ÷ 2 = 1 r 1, 1 ÷ 2 = 0 r 1: 1101₂

8. 8Hexadecimal numbersA to F are 10 to 15; place values 1, 16, 256

   Method2F₁₆ = 2 × 16 + 15 = 47

   Another way: Binary in groups of four11001000₂: 1100 = 12 = C and 1000 = 8, so C8₁₆

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Multiplying and dividing powers →

   Simplify: 18x¹⁵ ÷ 6x⁵ = x^□

   Fill in each blank

   ▶Show the answer and working

   Answer: 3, 10

   Divide the numbers: 18 ÷ 6 = 3; subtract the indices: 15 − 5 = 10. So 3x¹⁰.

2. Example 2Writing powers with the same base →

   Solve: 27^x = 81, x =

   Answer in simplest form

   ▶Show the answer and working

   Answer: 43

   Both sides as powers of 3: 3^(3x) = 3⁴, so 3x = 4 and x = 43.

3. Example 3Hexadecimal numbers →

   CE₁₆ = × 16 + =

   Fill in each blank

   ▶Show the answer and working

   Answer: 12, 14, 206

   A = 10, B = 11, C = 12, D = 13, E = 14, F = 15: 12 × 16 + 14 = 206.