Indices, surds and standard form
Maths · GCSE · The School
The index laws
Multiplying powers of the same base adds the indices; dividing subtracts them; a power of a power multiplies them. Anything to the power zero is 1. A negative index means one over the positive version. A fractional index is a root: to the power one half is the square root, and to the power two thirds is the cube root, then squared.
Surds
A surd is a root left exact rather than rounded, and leaving it exact is usually the point. Simplify by pulling out square factors: root 50 is root 25 times root 2, which is 5 root 2. To rationalise a denominator, multiply top and bottom by the surd — 1 over root 3 becomes root 3 over 3. Note that root a plus root b is NOT root of a plus b; testing with 9 and 16 shows why in one line.
Standard form
A number in standard form is written as a value between 1 and 10 multiplied by a power of ten. Multiplying means multiplying the front numbers and adding the powers; dividing means dividing and subtracting. Adding and subtracting need the same power of ten first, which is the step people skip. Check the front number is still between 1 and 10 at the end, and adjust if it is not.
Simplify (2 × 10³) × (4 × 10⁻⁵). FRONT NUMBERS: 2 × 4 = 8. POWERS: add the indices, 3 + (−5) = −2. So 8 × 10⁻². Is the front number between 1 and 10? Yes, 8 is, so it is already in standard form. As an ordinary number that is 0.08 — and notice the negative index gave a small POSITIVE number, not a negative one. NOW A HARDER ONE: (6 × 10⁴) ÷ (3 × 10⁻²). Divide the fronts, 6 ÷ 3 = 2. Subtract the powers, 4 − (−2) = 6. So 2 × 10⁶. Subtracting a negative caught you if you got 10², which is the single commonest slip on this topic.