The Basics

Indices indicate that a number should be multiplied by itself, with the index (the number written in superscript) referring to the amount of times it should be multiplied by itself. For example, when given the calculation:

2^{4}

you would expand it to:

2 \times 2 \times 2 \times 2

You don’t use the index as the amount of times it is being multiplied, rather as the amount of times the number occurs in the calculation.

Addition and Subtraction

Numbers with powers cannot be added to each other or subtracted from each other without calculating what each power means first.

Multiplication and Division

When using indices in Algebra, there may be a need to multiply two indices together. To do this, you add the index of the first number to the index of the second number. In other words:

x^{a} \times x^{b} = x^{a + b}

When you need to divide two indices, a similar idea has to be used, instead subtracting the second index, like this:

x^{a} \div x^{b} = x^{a - b}

Powers to the Power of a Power

When a power is raised to a power, you must multiply the powers together. This is shown as:

(x^{a})^{b} = x^{ab}

Powers of 0

When a number is to the power of zero, the result is always 1.

Negative Indices

Negative Indices mean that the reciprocal of the number must be found. To find the reciprocal, you would convert $x$ to $\frac{1}{x}$ , meaning that:

x^{- 1} = \frac{1}{x}

However, when a number is raised to a power less than -1, it must be first raised to the positive power before finding the reciprocal, so:

x^{- y} = \frac{1}{x ^{y}}

Fractional Indices

To solve a fractional index, you must first root by the denominator, before raising the result to the numerator. This is the formula:

x^{\frac{a}{b}} = (b x)^{a}

Revision

Indices