# 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 ; $\sqrt{4.9}=\sqrt{4}.\sqrt{9}$.

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 $\sqrt[n]{ab}=\sqrt[n]{a}.\sqrt[n]{b}$

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 : $\sqrt[n]{a}.\sqrt[n]{b}=x.y=\sqrt[n]{(xy)^n}=\sqrt[n]{x^ny^n}=\sqrt[n]{ab}$

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 : $\sqrt{75}=\sqrt{25.3}=\sqrt{25}.\sqrt{3}=5\sqrt{3}$

Notice that $5.\sqrt{3}=5\sqrt{3}$

Example 1 # :

Simplify $\sqrt{700}$

Solution : $\sqrt{700}=\sqrt{100.7}=\sqrt{100}.\sqrt{7}=10\sqrt{7}$

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. $\sqrt{700}=10\sqrt{7}=10.(2.646)=26.46$

Example 2 # :

Simplify $5\sqrt{18}$

Solution : $5\sqrt{18}=5\sqrt{9.2}$ $5\sqrt{18}=5\sqrt{9}.\sqrt{2}$ $5\sqrt{18}=5.3\sqrt{2}$ $5\sqrt{18}=15\sqrt{2}$

Example 3 # :

Simplify $\sqrt{250}$

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

Therefore, $\sqrt{250}=\sqrt{2.5^3}$ $\sqrt{250}=\sqrt{2}.\sqrt{5^3}$ $\sqrt{250}=5\sqrt{2}$

Example 4# :

Simplify $\sqrt{288x^5}$

Solution :

288 = 2. 2. 2. 2. 2. 3. 3 $x^5=x^4.x$ $\sqrt{288x^5}=\sqrt{2^4.2.3^2.x^4.x}$ $=\sqrt{2^4.3^2.x^4}.\sqrt{2.x}$ $=2^2.3.x^2.\sqrt{2.x}$ $=12x^2\sqrt{2x}$