# The first law of Radicals

First law of radicals

Consider the expression . We know that . Also, . Therefore we can write ;

.

Also Read:

This example illustrates an important property of radicals. The property states that a radical sign may be distributed over a product provided that each factor is positive, and it is true for every index n.

**First law of radicals**

For all positive number *a* and b.

The first law of radicals can be proved as follows:

Let . Then x is an nth root of *a *and y is an nth root of b. By the definition of nth root we can write that . Therefore we have :

Which is the desired result.

We may use the first law of radicals to simplify certain types of radical expressions. For example, to simplify we find the largest factor of 75 which is perfect square and then apply the first law of radicals. Our work looks like this :

Notice that

**Example 1 # :**

Simplify

Solution :

Using this last form it is an easy matter to approximate . The number 700 is too large to be found in our table of square roots, so we find the square root of 7 and multiply by 10.

**Also Read : roots and radicals**

**Example 2 # :**

Simplify

Solution :

**Example 3 # :**

Simplify

Solution :

In this case we look for the largest perfect cube that is a factor of 250. The procedure that was used to factor large integers. This procedure is illustrated below :

250 = 2. 5. 5. 5

Therefore,

**Example 4# :**

Simplify

Solution :

288 = 2. 2. 2. 2. 2. 3. 3