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 2^{3}, means 2 multiplied by itself three times: 2 × 2 × 2 = 8. Similarly, x raised to the power of 4, written as x^{4}, 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:

**Multiplying powers with the same base:**add the indices, e.g., 2

^{3}× 2

^{4}= 2

^{(3+4)}= 2

^{7}

**Dividing powers with the same base:**subtract the indices, e.g., 2

^{5}÷ 2

^{3}= 2

^{(5-3)}= 2

^{2}

**Raising a power to another power:**multiply the indices, e.g., (2

^{3})

^{2}= 2

^{(3×2)}= 2

^{6}

**Powers of 0 and 1:**0 raised to any positive power is 0, and 1 raised to any power is 1, e.g., 0

^{3}= 0, 1

^{10}= 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 2^{0} = 1, x^{0} = 1, or a^{0} = 1.

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