Express four root two multiplied by root 48 in its simplest form.
So we’ve been given a product of two surds. In order to express this in its simplest form, we need to evaluate the product, but also write it in a form involving surds, which cannot be simplified further.
Let’s consider root 48 first. In order to simplify a surd, we look for square factors. You will recall that 48 is equal to 16 multiplied by three and 16 is a square number. It’s equal to four squared. Therefore, we can simplify the square root of 48 by first writing it as the square root of 16 multiplied by three.
A key factor that we’ll use within this question tells us how to deal with the square root of a product. The square root of the product 𝑎𝑏 can in fact be written as the product of the individual square roots. The square root of 𝑎𝑏 is equal to the square root of 𝑎 multiplied by the square root of 𝑏.
The square root of 48 can therefore be separated out into the square root of 16 multiplied by the square root of three. 16 remember is a squared number and so its square root is an integer. It’s four. The square root of 48 is therefore equal to four multiplied by the square root of three or four root three. So by looking for square factors of 48, we’ve simplified part of the expression. Root 48 is equivalent to four root three.
Now, let’s substitute this back into the product we were asked to find. Four root two multiplied by root 48 is therefore equal to four root two multiplied by four root three. In order to evaluate this product, we recall the fact that multiplication is commutative. It doesn’t matter which order we multiply the different parts together in. This product is equal to four multiplied by root two multiplied by four multiplied by root three. And we can multiply in whichever order we like. If we multiply the integers first of all, then we have four multiplied by four which is 16.
Next, let’s consider the two surds: root two multiplied by root three. We can apply the same results as earlier, but in reverse this time. The square root of two multiplied by the square root of three is equal to the square root of two times three. Two times three is equal to six. Therefore, the product becomes 16 root six.
This surd can’t be simplified any further as six doesn’t have any square factors. Therefore, this answer is in its simplest form. Four root two multiplied by root 48 is equal to 16 root six.