Welcome to this deck on the laws of indices. Indices, also known as powers or exponents, are a shorthand way of writing repeated multiplication. For example, instead of writing 5 times 5 times 5, we can write 5 to the power of 3. In this expression, 5 is the 'base' and 3 is the 'index'. In this deck, we'll learn the rules for combining these powers when we multiply, divide, or raise a power to another power. We'll also cover what zero, negative, and fractional indices mean.

Let's start with the multiplication law. This rule applies when you multiply two powers that have the same base. For example, consider 2 to the power of 3, multiplied by 2 to the power of 2. This is just (2 times 2 times 2) multiplied by (2 times 2). In total, we are multiplying five 2s together, which is 2 to the power of 5. Notice that the new index, 5, is simply the original indices, 3 and 2, added together. This leads to our first rule.

Next is the division law, which works in a similar way. Imagine we want to calculate 2 to the power of 5, divided by 2 to the power of 2. We can write this as a fraction. On top, we have five 2s multiplied together, and on the bottom, we have two 2s. We can cancel two of the 2s from the top and bottom, leaving us with three 2s on top. So the answer is 2 to the power of 3. Notice that the new index, 3, is the result of subtracting the original indices: 5 minus 2. This gives us our second rule.