Roots and Radical Notation

We know what an exponent is, right? You multiply a number by itself a certain number of times, and we know the notation for exponents. The four is the base number, and the two is the exponent. This is four multiplied by itself. But what if we want to sort of do this in reverse? Say you have a budget of $50 for a square patio by your back door. Each tile costs $2, so you can afford to buy 25 tiles. How big of a patio can you afford to make? Since it's a square patio, the width and the length will be the same. We know that the area of a square is the length of one of its sides squared. If we remember multiplication tables, we know that five times five equals 25, and five times five is five squared. So that's the answer. We can use our 25 tiles to build a five foot by five foot patio. We won't be throwing any large parties on our patio, but we can put a nice deck chair out there. But let's say we know a particular number. What do we call a number that when multiplied by itself gives us that particular number. This number is called a square root. We call this a square root because we are finding the length of one of the sides of a given square area. And a square is a type of exponent. But what about other types of exponents, like a cube. Remember that a cube is an exponent of three. Can we do a cube root? Find out how wide a box we need to make in order to hold 27 cubic feet of space? Yes, we can. The times three times three is 27. So the cube root of 27 is three. If we need a cubic box to hold 27 cubic feet, it will be three feet on a side. So square roots and cube roots are useful stuff? They can help us solve problems with covering surfaces and filling containers. But what about other exponents? Can we do roots for them? Is there a sixth root? A seventh root, a 235th root? Yes. And we can use a special notation to make expressions with these kinds of roots. This symbol is called a radical. We use it like this. The represents the number of times we multiply the root by itself to get x. In this notation, the n is called the index of the radical. The X is the number we are taking the root of, and we call it the radicand. Here, we replace the n with a five and the x with 32. So this is the fifth root of 32, which is equal to two, and two to the fifth power is 32. We can see the relationship between the radical notation and the exponent notation. The base number and the exponent notation is the answer to the radical notation. The radicand in the radical notation is the answer to the exponent notation. And the index in the radical notation is the exponent in the exponent notation. Working with roots and radical notation is a complex topic, and there's plenty more to learn about them. But this should help you get started understanding them and adding them to your math skills.