Indices and roots are fundamental concepts in mathematics, especially in algebra. In this context, an index refers to the power or exponent of a number, while a root relates to the inverse operation of exponentiation. In this explanation, we'll focus on using indices and roots in GCSE mathematics.

Indices, or powers, are a shorthand way of showing that a number or variable has been multiplied by itself a certain number of times. For example, 2 raised to the power of 3, written as 23, means 2 multiplied by itself three times: 2 × 2 × 2 = 8. Similarly, x raised to the power of 4, written as x4, means x multiplied by itself four times: x × x × x × x.

We can also have negative and fractional indices. A negative index means the number is in the denominator instead of the numerator, and the exponent is positive. For example, 2 raised to the power of -2, written as 2-2, means 1 divided by 2 multiplied by itself twice: 1/(2 × 2) = 1/4.

A fractional index means that the root of the number is taken, where the fraction's denominator is the root's power. For example, the cube root of 27, written as 273, means finding a number that, when multiplied by itself three times, gives you 27. This number is 3 since 3 × 3 × 3 = 27.

It's essential to learn the laws of indices to help perform calculations involving powers and roots. The primary rules are:

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Multiplying powers with the same base: add the indices, e.g., 23 × 24 = 2(3+4) = 27
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Dividing powers with the same base: subtract the indices, e.g., 25 ÷ 23 = 2(5-3) = 22
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Raising a power to another power: multiply the indices, e.g., (23)2 = 2(3×2) = 26
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Powers of 0 and 1: 0 raised to any positive power is 0, and 1 raised to any power is 1, e.g., 03 = 0, 110 = 1

One crucial fact to remember is that anything to the power of 0 always equals 1. This means that any number or variable raised to the power of 0 is 1, such as 20 = 1, x0 = 1, or a0 = 1.