How do you find a square root of a number that's not a perfect square? I'm very confused. The book doesn't explain it too well.

You can approximate it or simplify it in terms of (products of) square roots of smaller numbers.

E.g. consider sqrt[117]

The number 177 can be factored:

117 = 9*13

So we have:

sqrt[177] = sqrt[9*13] =
sqrt[9]*sqrt[13] = 3 sqrt[13]

Cases like:

sqrt[a]/sqrt[b]

can be rewritten by multiplying numerator and denominator by sqrt[b]:

sqrt[a]/sqrt[b] =

sqrt[a]*sqrt[b]/(sqrt[b]*sqrt[b]) =

sqrt[a]*sqrt[b]/b

To find the square root of a number that is not a perfect square, one approach is to approximate it using a calculator or computer program that has a square root function. This will give you an approximate value for the square root.

Another approach is to simplify the square root in terms of products of square roots of smaller numbers. Let's go through an example to illustrate this method.

Let's say we want to find the square root of 117. We can start by factoring 117 into its prime factors, which are 9 and 13:

117 = 9 * 13

Next, we can rewrite the square root using the property that the square root of a product is equal to the product of the square roots:

√117 = √(9 * 13)

Taking the square root of 9 gives us 3:

√117 = 3 * √13

So the square root of 117 can be simplified as 3 times the square root of 13.

In cases where you have a fraction involving square roots, you can simplify it by multiplying the numerator and denominator by the square root of the denominator. For example, if you have the fraction √a/√b, you can rewrite it as:

√a/√b = (√a * √b) / (√b * √b)

This simplifies to:

√a/√b = √(a * b) / b

By simplifying square roots in this way, you can express them in terms of smaller square roots, making them easier to work with.