Mr Kidman's Classroom
Revise the key ideas
Powers, roots and index laws
- In 34, 3 is the base and 4 is the index: 3 × 3 × 3 × 3 = 81. Squaring and cubing mean powers 2 and 3. A power is not base × index.
- Square roots undo squaring: √49 = 7. The symbol √ gives the non-negative square root; the equation x2 = 49 has two solutions, x = ±7. Cube roots can be negative: ∛(−8) = −2.
- Recognise useful powers: 25 = 32, 33 = 27, 43 = 64 and 53 = 125. Know common squares and cubes so you can undo familiar powers without a calculator.
- With the same base, multiply by adding indices and divide by subtracting: aman = am+n; = am−n for a ≠ 0.
- A power of a power multiplies indices: (am)n = amn. Also (ab)n = anbn. You cannot apply these rules to sums: (a + b)2 is not a2 + b2.
- For a non-zero base, a0 = 1 and a−n = . For example, 2−3 = . A negative index does not make the value negative.
Standard form
- Standard form is A × 10n, where 1 ≤ A < 10 and n is an integer. 43 000 = 4.3 × 104; 0.00072 = 7.2 × 10−4.
- Positive powers of 10 describe large numbers; negative powers describe small positive numbers. 0.003 is 3 × 10−3, not −3000.
- Worked example: (3 × 105)(4 × 10−2) = 12 × 103 = 1.2 × 104. Multiply the coefficients (the numbers in front of the powers of 10), add indices, then adjust the coefficient to the required range.
- Worked example: = 3 × 104. Divide coefficients and subtract indices.
The coefficient is 6 ÷ 2 = 3. The power is 10 to the index 7 − 3 = 4, giving 3 × 104.
- For addition/subtraction, rewrite with the same power of 10: 3.2 × 104 + 6 × 103 = (3.2 + 0.6) × 104 = 3.8 × 104.
- On a calculator, use its standard-form entry key and brackets around complete numbers when dividing. Check whether the displayed exponent is positive or negative.
Higher — fractional indices
- Estimate roots using nearby known powers: since 82 < 70 < 92, √70 lies between 8 and 9. Trial values can refine an estimate but an estimate does not prove the root is rational (exactly expressible as a fraction of integers).
- For a positive base, a means the nth root of a, and a means raise the nth root to power m. Thus 27 = (∛27)2 = 9.
- Worked example: 16 = = = . Handle the negative sign in the index by taking a reciprocal.
- Fractional and negative indices obey the same index laws within their real-number domains. For example, x × x = x2 for x > 0.
The domain means the values allowed for the input. Here x > 0 ensures the square roots exist as real numbers, so these fractional powers can be combined in the stated way.
