# 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 $\sqrt[3]{250}$

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

Therefore,

$\sqrt[3]{250}=\sqrt[3]{2.5^3}$

$\sqrt[3]{250}=\sqrt[3]{2}.\sqrt[3]{5^3}$

$\sqrt[3]{250}=5\sqrt[3]{2}$

Example 4# :

Simplify $\sqrt{288x^5}$

Solution :

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

$x^5=x^4.x$

$\sqrt{288x^5}=\sqrt{2^4.2.3^2.x^4.x}$

$=\sqrt{2^4.3^2.x^4}.\sqrt{2.x}$

$=2^2.3.x^2.\sqrt{2.x}$

$=12x^2\sqrt{2x}$

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