The first law of Radicals

First law of radicals

Consider the expression \sqrt{4.9}. We know that \sqrt{4.9}=\sqrt{36}=6. Also, \sqrt{4}.\sqrt{9}=2.3=6. 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 x=\sqrt[n]{a}\text{ and }y=\sqrt[n]{b}. 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 x^n=a\text{ and }y^n=b. 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 \sqrt{75} 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 5.\sqrt{3}=5\sqrt{3}

Example 1 # :

Simplify \sqrt{700}

Solution :


Using this last form it is an easy matter to approximate \sqrt{700}. 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 5\sqrt{18}

Solution :





Example 3 # :

Simplify \sqrt[3]{250}

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





Example 4# :

Simplify \sqrt{288x^5}

Solution :

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






%d bloggers like this: