Laws of indices and square roots · Year 10
In Laws of indices and square roots (Year 10) you practise 3 skills: Multiplying and dividing powers of the same base · A power of a power and a power of a product · The index 0 and the sign of a power. A power is repeated multiplication of one number: the base is the number multiplied, and the index says how many times. When you multiply two powers with the same base, the factors join into one row, so add the indices. When you divide, the shared factors cancel, so subtract the indices.
For a power of a power, multiply the indices; for a power of a product, raise each factor separately. Any number other than 0 to the power 0 equals 1. When a negative base is in brackets, an even index gives a positive result and an odd index a negative one. The square root of a product equals the product of the square roots, and the square root of a quotient equals the quotient of the square roots. Addition has no such law: the square root of a sum is not the sum of the square roots. The square root of a number squared is the number without its sign, that is, its absolute value.
Tip: Before using a law, check that the bases are the same. Powers with different bases have no law for adding the indices.
What you learn
Multiplying and dividing powers of the same base
For example: Write as a single power: 3⁷ × 3²
Answer and solution
The answer: 9
Add the indices: 7 + 2 = 9 . So 3⁷ × 3² = 3⁹
A power of a power and a power of a product
For example: Write as a single power: (3²)⁴
Answer and solution
The answer: 8
(3²)⁴ = 3² × 3² × 3² × 3² . Multiply the indices: 2 × 4 = 8 . So (3²)⁴ = 3⁸
The index 0 and the sign of a power
For example: Work out: x⁰ when x ≠ 0
Answer and solution
The answer: 1
x⁰ = 1
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).